The ideal-valued index for a dihedral group action, and mass partition
  by two hyperplanes
The ideal-valued index for a dihedral group action,
and mass partition by two hyperplanes
Pavle V. M. Blagojević∗
Mathematički Institut
Knez Michailova 35/1
11001 Beograd, Serbia
pavleb@mi.sanu.ac.rs
Günter M. Ziegler∗∗
Inst. Mathematics, MA 6-2
TU Berlin
D-10623 Berlin, Germany
ziegler@math.tu-berlin.de
December 9, 2010
Abstract
We compute the complete Fadell–Husseini index of the dihedral group D8 = (Z2)
2 ⋊ Z2 acting
on Sd × Sd for F2 and for Z coefficients, that is, the kernels of the maps in equivariant cohomology
(pt,F2) −→ H
(pt,Z) −→ H∗D8(S
This establishes the complete cohomological lower bounds, with F2 and with Z coefficients, for the
two-hyperplane case of Grünbaum’s 1960 mass partition problem: For which d and j can any j
arbitrary measures be cut into four equal parts each by two suitably chosen hyperplanes in Rd? In
both cases, we find that the ideal bounds are not stronger than previously established bounds based
on one of the maximal abelian subgroups of D8.
Contents
1 Introduction 2
1.1 The hyperplane mass partition problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Statement of the main result (k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Proof overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Evaluation of the index bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 F2-evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.2 Z-evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Configuration space/Test map scheme 8
2.1 Configuration space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Test map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 The test space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
∗The research leading to these results has received funding from the European Research Council under the European
Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels. Also supported
by the grant ON 174008 of the Serbian Ministry of Science and Environment.
∗∗The research leading to these results has received funding from the European Research Council under the European
Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels.
http://arxiv.org/abs/0704.1943v4
pavleb@mi.sanu.ac.rs
ziegler@math.tu-berlin.de
3 The Fadell–Husseini index theory 11
3.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 IndexG,R and Index
G,R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 The restriction map and the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Basic calculations of the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4.1 The index of a product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4.2 The index of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 The cohomology of D8 and the restriction diagram 17
4.1 The poset of subgroups of D8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 The cohomology ring H∗(D8,F2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 The cohomology diagram of subgroups with coefficients in F2 . . . . . . . . . . . . . . . . 19
4.3.1 The Z2 × Z2-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.2 The D8-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 The cohomology ring H∗(D8,Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4.1 Evens’ view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4.2 The Bockstein spectral sequence view . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5 The D8-diagram with coefficients in Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5.1 The Z2 × Z2-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.5.2 The D8-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 IndexD8,F2S(R
4 ) 27
5.1 IndexD8,F2S(V−+ ⊕ V+−) = 〈w〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 IndexD8,F2S(V−−) = 〈y〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.3 IndexD8,F2S(R
4 ) = 〈y
jwj〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 IndexD8,ZS(R
4 ) 28
6.1 The case when j is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 The case when j is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 IndexD8,F2S
d×Sd 33
7.1 The d-th row as an H∗(D8,F2)-module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 Indexd+2D8,F2S
d × Sd = 〈πd+1, πd+2〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.3 IndexD8,F2S
d × Sd = 〈πd+1, πd+2, w
d+1〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.4 An alternative proof, sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 IndexD8,ZS
d×Sd 39
8.1 The d-th row as an H∗(D8,Z)-module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.1.1 The case when d is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.1.2 The case when d is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
8.2 Indexd+2D8,ZS
d × Sd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8.2.1 The case when d is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8.2.2 The case when d is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1 Introduction
1.1 The hyperplane mass partition problem
A mass distribution on Rd is a finite Borel measure µ(X) =
fdµ determined by an integrable density
function f : Rd → R.
Every affine hyperplane H = {x ∈ Rd | 〈x, v〉 = α} in Rd determines two open halfspaces
H− = {x ∈ Rd | 〈x, v〉 < α} and H+ = {x ∈ Rd | 〈x, v〉 > α}.
An orthant of an arrangement of k hyperplanesH = {H1, H2, . . . , Hk} in R
d is an intersection of halfspaces
O = Hα11 ∩ · · · ∩ H
k , for some αj ∈ Z2. Thus there are 2
k orthants determined by H and they are
naturally indexed by elements of the group (Z2)
An arrangement of hyperplanes H equiparts a collection of mass distributions M in Rd if for each
orthant O and each measure µ ∈ M we have
µ(O) = 1
µ(Rd).
A triple of integers (d, j, k) is admissible if for every collectionM of j mass distributions in Rd there
exists an arrangement of k hyperplanes H equiparting them.
The general problem formulated by Grünbaum [16] in 1960 can be stated as follows.
Problem 1.1. Determine the function ∆ : N2 → N given by
∆(j, k) = min{d | (d, j, k) is an admissible triple}.
The case of one hyperplane, ∆(j, 1) = j, is the famous ham sandwich theorem, which is equivalent
to the Borsuk–Ulam theorem. The equality ∆(2, 2) = 3, and consequently ∆(1, 3) = 3, was proven by
Hadwiger [17]. Ramos [30] gave a general lower bound for the function ∆,
∆(j, k) ≥ 2
j. (1)
Recently, Mani-Levitska, Vrećica and Živaljević [26] applied Fadell–Husseini index theory for an elemen-
tary abelian subgroup (Z2)
k of the Weyl group Wk = (Z2)
k ⋊ Sk to obtain a new upper bound for the
function ∆,
∆(2q + r, k) ≤ 2k+q−1 + r. (2)
In the case of j = 2l+1 − 1 measures and k = 2 hyperplanes these bounds yield the equality
∆(j, 2) = ⌈ 3
1.2 Statement of the main result (k = 2)
This paper addresses Problem 1.1 for k = 2 using two different but related Configuration Space/Test
Map schemes (Section 2, Proposition 2.2).
• The product scheme is the classical one, already considered in [32] and [26]. The problem is
translated to the problem of the existence of a Wk-equivariant map,
Yd,k :=
(R2k)
where Wk = (Z2)
k ⋊ Sk is the Weyl group.
• The join scheme is a new one. It connects the problem with classical Borsuk–Ulam properties in
the spirit of Marzantowicz [27]. It asks the question whether there exists a Wk-equivariant map
Xd,k :=
Uk × (R2k)
The Wk-representations R2k and Uk are introduced in Section 2.2.
Obstruction theory methods cannot be applied to either scheme directly for k > 1, since the Wk-actions
on the respective configuration spaces
are not free (compare [26, Section 2.3.3],
assumptions on the manifold Mn). Therefore we analyze the associated equivariant question for k = 2
via the Fadell–Husseini ideal index theory method. We show that the join scheme considered from the
Fadell–Husseini point of view, with either F2 or Z coefficients, yields no obstruction to the existence of
the equivariant map in question (Remarks 5.3 and 6.3). In the case of the product scheme we give the
ideal bounds obtained from the use of the full group of symmetries by proving the following theorem.
Theorem 1.2. Let πd, d ≥ 0, be polynomials in F2[y, w] given by
πd(y, w) =
d− 1− i
wiyd−2i
and Πd, d ≥ 0, be polynomials in Z[Y,M,W ]/〈2Y, 2M, 4W ,M
2 −WY〉 given by
Πd(Y,W) =
d− 1− i
W iYd−2i.
(A) F2-bound: The triple (d, j, 2) ∈ N
3 is admissible if
yjwj /∈ 〈πd+1, πd+2〉 ⊆ F2[y, w].
(B) Z-bound: The triple (d, j, 2) ∈ N3 is admissible if
(j − 1)mod2 Y
jmod2 Y
2 M, jmod 2 Y
〈 (d− 1)mod 2 Π d+2
, (d− 1)mod2 Π d+4
(d− 1)mod 2 MΠ d
dmod 2 Π d+1
, dmod 2 Π d+3
in the ring Z[Y,M,W ]/〈2Y, 2M, 4W ,M2 −WY〉.
Remark 1.3. Let Π̂d, d ≥ 0, be the sequence of polynomials in Z[Y,W ] defined by Π̂0 = 0, Π̂1 = Y
and Π̂d+1 = Y Π̂d +W Π̂d−1 for d ≥ 2. Then the sequences of polynomials Πd and πd are reductions of
the polynomials Π̂d. The polynomials Π̂d can be also described by the generating function (formal power
series)
Π̂d =
1− Y −W
where Π̂d is homogeneous of degree 2d if we set deg(Y ) = 2 and deg (W ) = 4.
Theorem 1.2 is a consequence of a topological result, the complete and explicit computation of the
relevant Fadell–Husseini indexes of the D8-space S
d × Sd and the D8-sphere S(R
Theorem 1.4.
(A) Index
D8,F2
4 ) = IndexD8,F2S(R
4 ) = 〈y
jwj〉.
(B) Indexd+2D8,F2(S
d × Sd) = 〈πd+1, πd+2〉.
(C) Index
4 ) =
2 〉 , for j even,
2 M,Y
2 〉 , for j odd.
(D) Indexd+2D8,ZS
d × Sd =
〈Π d+2
,Π d+4
,MΠ d
〉 , for d even,
〈Π d+1
,Π d+3
〉 , for d odd.
The sequence of Fadell–Husseini indexes will be introduced in Section 3. The actions of the dihedral
group D8 and the definition of the representation space R
4 are given in Section 2. Even though it does
not seem to have any relevance to our study of Problem 1.1, the complete index IndexD8,F2(S
d×Sd) will
also be computed in the case of F2 coefficients,
IndexD8,F2(S
d × Sd) = 〈πd+1, πd+2, w
d+1〉. (3)
Final remark 1.5. The preprint versions of this paper, posted on the arXiv in April 2007 and July
2008, arXiv0704.1943v1–v2, have been referenced in diverse applications: see Gonzalez and Landweber
[15], Adem and Reichstein [2], as well as [6].
1.3 Proof overview
The Problem 1.1 about mass partitions by hyperplanes can be connected with the problem of the existence
of equivariant maps as discussed in Section 2, Proposition 2.2. The topological problems we face, about
the existence of Wk = (Z2)
k ⋊ Sk-equivariant maps, for the product / join schemes,
Uk ×R
have to be treated with care because the actions of the Weyl groups Wk are not free. Note that there is
no naive Borsuk–Ulam theorem for fixed point free actions. Indeed, in the case k = 2 when W2 = D8
there exists a W2-equivariant map [5, Theorem 3.22, page 49]
(V+− ⊕ V−+)
(U2 ⊕ V−−)
even though dim (V+− ⊕ V−+)
> dim (U2 ⊕ V−−)
. The W2 = D8-representations V+− ⊕ V−+, V−−
and U2 are introduced in Section 2.2.
In this paper we focus on the case of k = 2 hyperplanes. Theorem 1.2 gives the best possible answer
to the question about the existence of W2 = D8-equivariant maps
Sd × Sd −→ S(R
from the point of view of Fadell–Husseini index theory (Section 3). We explicitly compute the relevant
Fadell–Husseini indexes with F2 and Z coefficients (Theorem 1.4, Sections 5, 6, 7 and 8). Then Theorem
1.2 is a consequence of the basic index property, Proposition 3.2.
The index of the sphere S(R
4 ), with F2 coefficients, is computed in Section 5 by
• decomposition of the D8-representation R
4 into a sum of irreducible ones, and
• computation of indexes of spheres of all irreducible D8-representations.
The main technical tool is the restriction diagram derived in Section 4.3.2, which connects the indexes of
the subgroups of D8.
The index with Z coefficients is computed in Section 6 using
• (for j even) the results for F2 coefficients and comparison of Serre spectral sequences, and
• (for j odd) the Bockstein spectral sequence combined with known results for F2 coefficients and
comparison of Serre spectral sequences.
The index of the product Sd × Sd is computed in Sections 7 and 8 by an explicit study of the Serre
spectral sequence associated with the fibration
Sd × Sd → ED8 ×D8 (S
d × Sd)→ BD8.
The major difficulty comes from non-triviality of the local coefficients in the Serre spectral sequence. The
computation of the spectral sequence with non-trivial local coefficients is done by an independent study
of H∗(D8,F2)-module and H
∗(D8,Z)-module structures of relevant rows in the Serre spectral sequence
(Sections 7.1 and 8.1).
1.4 Evaluation of the index bounds
1.4.1 F2-evaluation
It was pointed out to us by Sinǐsa Vrećica that, with F2-coefficients, the D8 index bound gives the same
bounds as the H1 = (Z2)
index bound. This observation follows from the implication
ajbj(a+ b)j ∈ 〈ad+1, (a+ b)d+1〉 ⇒ ajbj(a+ b)j ∈ 〈ad+1 + (a+ b)d+1, ad+2 + (a+ b)d+2〉.
By introducing a new variable c := a+ b, it is enough to prove the implication
ajcj(a+ c)j ∈ 〈ad+1, cd+1〉 ⇒ ajcj(a+ c)j ∈ 〈ad+1 + cd+1, ad+2 + cd+2〉. (4)
Let us assume that ajcj(a + c)j ∈ 〈ad+1, cd+1〉. The monomials in the expansion of ajcj(a + c)j always
come in pairs
ad+kc3j−d−k + cd+ka3j−d−k.
This is also true when j is even since
=mod2 0 implies there are no middle terms. The sequence of
equations
ad+1c3j−d−1 + cd+1a3j−d−1 = (ad+1 + cd+1)(c3j−d−1 + a3j−d−1) + a3j + c3j
ad+2c3j−d−2 + cd+2a3j−d−2 = (ad+1 + cd+1)(ac3j−d−2 + a3j−d−2c) + a3j−1c+ ac3j−1
. . . .
a3j + c3j = (ad+2 + cd+2)(a3j−d−2 + c3j−d−2) + ad+2c3j−d−2 + cd+2a3j−d−2
shows that all the binomials
ad+1c3j−d−1 + cd+1a3j−d−1, ad+2c3j−d−2 + cd+2a3j−d−2, . . . , a3j + c3j
belong to the ideal 〈ad+1 + cd+1, ad+2 + cd+2〉 or none of them do.
Since for 3j − d− 1 even
3j−d−1
3j−d−1
2 + c
3j−d−1
3j−d−1
2 = (ad+1 + cd+1)a
3j−d−1
3j−d−1
∈ 〈ad+1 + cd+1, ad+2 + cd+2〉
and for 3j − d− 1 odd
ad+2+
3j−d−2
3j−d−2
2 + cd+2+
3j−d−2
3j−d−2
2 = (ad+2 + cd+2)a
3j−d−2
3j−d−2
∈ 〈ad+1 + cd+1, ad+2 + cd+2〉
the implication (4) is proved.
1.4.2 Z-evaluation
More is true, even the complete D8 index bound, now with Z-coefficients, implies the same bounds as
does the subgroup H1 = (Z2)
for the k = 2 hyperplanes mass partition problem.
Lemma 1.6. Let a =
i and b =
i be the dyadic expansions. Then
mod 2
This classical fact [25] about binomial coefficients mod 2 yields the following property for the sequence
of polynomials Πd, d ≥ 0.
Lemma 1.7. Let q > 0 and i be integers. Then
2q−1−i
0, i 6= 0
1, i = 0
(B) Π2q = Y
Proof. The statement (B) is a direct consequence of the fact (A) and the definition of polynomials Πd.
For i /∈ {1, . . . , 2q−1} the statement (A) is true from boundary conditions on binomial coefficients. Let
i ∈ {1, . . . , 2q−1} and i =
k∈I⊆{0,...,q−1}
2k. Then
2q − 1− i = 20 + 21 + 22 + · · ·+ 2q−1 −
k∈I⊆{0,...,q−1}
k∈Ic⊆{0,...,q−1}
where Ic is the complementary index set in {0, . . . , q−1}. The statement (A) follows from Lemma 1.6
Let j be an integer such that j = 2q + r where 0 ≤ r < 2q and d = 2q+1 + r − 1. Let us introduce the
following ideals
2 〉, for j even,
2 M,Y
2 〉, for j odd,
and Bd =
〈Π d+2
,Π d+4
,MΠ d
〉, for d even,
〈Π d+1
,Π d+3
〉, for d odd.
The fact that theD8 index bound with Z-coefficients does not improve the mass partition bounds obtained
by using the subgroup H1 = (Z2)
is a consequence of the following facts:
• r = 0 ⇒ Aj ⊆ Bd,
• (r 6= 2q − 1 and Aj ⊆ Bd) =⇒ Aj+1 ⊆ Bd+1,
that are proved in Lemma 1.8 and 1.9, respectively.
Lemma 1.8. 〈Y2
〉 = A2q ⊆ B2q+1−1 = 〈Π2q ,Π2q+1〉.
Proof. Since Y2
= Π2q−1 by Lemma 1.7,
W = Π2q−1W = Π2q−1+2 + YΠ2q−1+1 ∈ 〈Π2q−1+1,Π2q−1+2〉.
By induction on the power i of W in Y2
W i ∈ 〈Π2q−1+i,Π2q−1+i+1〉,
and consequently
∈ 〈Π2q ,Π2q+1〉.
Lemma 1.9. If r 6= 2q − 1 and Aj ⊆ Bd then Aj+1 ⊆ Bd+1.
Proof. We distinguish two cases depending on the parity of j.
(A) Let j be even and Y
2 ∈ 〈Π d+1
,Π d+3
〉. There are polynomials α and β such that
2 = αΠ d+1
+ βΠ d+3
(j+1)+1
(j+1)−1
2 M = Y
2M = YM
αΠ d+1
+ βΠ d+3
∈ 〈Π (d+1)+2
,MΠ d+1
〉 ⊆ 〈Π d+3
,Π d+5
,MΠ d+1
〉 = Bd+1,
(j+1)+1
(j+1)+1
2 = YW
αΠ d+1
+ βΠ d+3
= αM2Π d+1
+ βYWΠ d+3
∈ 〈MΠ d+1
,Π d+3
〉 ⊆ 〈Π d+3
,Π d+5
,MΠ d+1
〉 = Bd+1.
Thus Aj+1 ⊆ Bd+1.
(B) Let j be odd and
2 M,Y
2 〉 = Aj ⊆ Bd = 〈Π d+2
,Π d+4
,MΠ d
There are polynomials α, β and γ such that
2 = αΠ d+2
+ βΠ d+4
+ γMΠ d
and no occurrence of the defining relation Π d+4
= YΠ d+2
+WΠ d
, Remark 1.3, can be subtracted
from the presentation. Then γMΠ d
∈ 〈Π d+2
,Π d+4
〉, and sinceM is of odd degree γ =Mγ′. In the
first case the inclusion Aj+1 ⊆ Bd+1 follows directly. Consider γ =Mγ
′. Since (Y+X )WΠi = YWΠi
for every i > 0, we have that
2 = αΠ d+2
+ βΠ d+4
+ γ′M2Π d
= αΠ d+2
+ βΠ d+4
+ γ′YWΠ d
= αΠ d+2
+ βΠ d+4
+ γ′Y(YΠ d
+1 +Π d
+2) ∈ 〈Π d+2
,Π d+4
〉 = Bd+1.
Thus Aj+1 ⊆ Bd+1.
Acknowledgements
We are grateful to Jon Carlson and to Carsten Schultz for useful comments and insightful observations.
The referee provided many useful suggestions and comments that are incorporated in the latest version
of the manuscript.
Some of this work was done in the framework of the MSRI program “Computational Applications of
Algebraic Topology” in the fall semester 2006.
2 Configuration space/Test map scheme
The Configuration Space/Test Map (CS/TM) paradigm (formalized by Živaljević in [31], and also beau-
tifully exposited by Matoušek in [28]) has been very powerful in the systematic derivation of topological
lower bounds for problems of Combinatorics and of Discrete Geometry.
In many instances, the problem suggests natural configuration spacesX , Y , a finite symmetry groupG,
and a test set Y0 ⊂ Y , where one would try to show that every G-equivariant map f : X → Y must
hit Y0. The canonical tool is then Dold’s theorem, which says that if the group actions are free, then the
map f must hit the test set Y0 ⊂ Y if the connectivity of X is higher than the dimension of Y \ Y0.
For the success of this “canonical approach” one crucially needs that a result such as Dold’s theorem
is applicable. Thus the group action must be free, so one often reduces the group action to a prime
order cyclic subgroup of the full symmetry group, and results may follow only in “the prime case” , or
with more effort and deeper tools in the prime power case. The main example for this is the Topological
Tverberg Problem, which is still not resolved for (d, q) if d > 1 and q is not a prime power [28, Section
6.4, page 165]. So in general one has to work much harder when the “canonical” approach fails.
In the following, we present configuration spaces and test maps for the mass partition problem.
2.1 Configuration space
The space of all oriented affine hyperplanes in Rd can be naturally identified with the subspace of the
sphere Sd obtained by removing two points, namely the “oriented hyperplanes at infinity”. Indeed, let
Rd be embedded in Rd+1 by (x1, . . . , xd) 7−→ (x1, . . . , xd, 1). Then every oriented affine hyperplane H in
Rd determines a unique oriented hyperplane H̃ through the origin in Rd+1 such that H̃ ∩ Rd = H , and
conversely if the hyperplane at infinity is included. The oriented hyperplane uniquely determined by the
unit vector v ∈ Sd is denoted by Hv and the assumed orientation is determined by the half-space H
Then H−−v = H
v . The obvious and classically used candidate for the configuration space associated with
the problem of testing admissibility of (d, j, k) is
Yd,k =
The relevant group acting on this space is the Weyl group Wk = (Z2)
k ⋊ Sk. Each Z2 = ({+1,−1}, ·)
acts antipodally on the appropriate copy of Sd (changing the orientation of hyperplanes), while Sk acts
by permuting copies. The second configuration space that we can use is
Xd,k = S
d ∗ · · · ∗ Sd
︸ ︷︷ ︸
k copies
∼= Sdk+k−1.
The elements of Xd,k are denoted by t1v1 + · · ·+ tkvk, with ti ≥ 0,
t1 = 1, vi ∈ S
d. The Weyl group
Wk acts on Xd,k by
εi · (t1v1 + · · ·+ tivi + · · ·+ tkvk) = t1v1 + · · ·+ ti(−vi) + · · ·+ tkvk,
π · (t1v1 + · · ·+ tivi + · · ·+ tkvk) = tπ−1(1)vπ−1(1) + · · ·+ tπ−1(i)vπ−1(i) + · · ·+ tπ−1(k)vπ−1(k),
where εi is the generator of the i-th copy of Z2 and π ∈ Sk is an arbitrary permutation.
2.2 Test map
LetM = {µ1, . . . , µj} be a collection of mass distributions in R
d. Let the coordinates of R2
be indexed
by the elements of the group (Z2)
k. The Weyl group Wk acts on R
2k by acting on its coordinate index
set (Z2)
k in the following way:
((β1, . . . , βk)⋊ π) · (α1, . . . , αk) =
β1απ−1(1), . . . , βkαπ−1(k)
The test map φ : Yd,k → (R
2k)j used with the configuration space Yd,k is a Wk-equivariant map given by
φ (v1, . . . , vk) =
∩ · · · ∩Hαkvk )−
(α1,...,αk)∈(Z2)k
i∈{1,...,j}
Denote the i-th component of φ by φi, i = 1, . . . , j.
To define a test map associated with the configuration space Xd,k, we discuss the (Z2)
k- and Wk-
module structures on R2
All irreducible representations of the group (Z2)
k are 1-dimensional. They are in bijection with the
homomorphisms (characters) χ : (Z2)
k → Z2. These homomorphisms are completely determined by the
values on generators ε1,. . . ,εk of (Z2)
k, i.e. by the vector (χ(ε1), . . . , χ(εk)). For (α1, . . . , αk) ∈ (Z2)
k let
Vα1...αk = span{vα1...αk} ⊂ R
2k denote the 1-dimensional representation given by
εi · vα1...αk = αi vα1...αk
The vector vα1...αk ∈ {+1,−1}
2k is uniquely determined up to a scalar multiplication by −1. Note that
〈vα1...αk , vβ1...βk〉 = 0
for α1 . . . αk 6= β1 . . . βk. For k = 2, with the abbreviation + for +1, − for −1, the coordinate index set
for R4 is {++,+−,−+,−−}. Then
v++ = (1, 1, 1, 1) , v+− = (1,−1, 1,−1),
v−+ = (1, 1,−1,−1) , v−− = (1,−1,−1, 1).
The following decomposition of (Z2)
k-modules holds, with the index identification (Z2)
k = {+,−}k,
k ∼= V+···+ ⊕
α1...αk∈(Z2)k\{+···+}
Vα1...αk
where V+···+ is the trivial (Z2)
k-representation. Let R2k denote the orthogonal complement of V+···+ and
π : R2
→ R2k the associated (equivariant) projection. Explicitly
R2k = {(x1, . . . , x2k) ∈ R
xi = 0} =
α1...αk∈(Z2)k\{+···+)}
Vα1...αk , (5)
x = (x1, . . . , x2k)
7−→ 1
〈x, vα1...αk)〉
α1...αk∈(Z2)k\{+···+}
where 〈·, ·〉 denotes the standard inner product of R2
. Observe that
im φ = φ(Yd,k) ⊆ (R2k)
Let α1 . . . αk ∈ (Z2)
k and let η(α1 . . . αk) =
αi). The following decomposition of Wk-modules
holds
k ∼= V+···+ ⊕
n=η(α1,...,αk)
Vα1...αk
∼= V+···+ ⊕R2k . (6)
The test map τ : Xd,k → Uk × (R2k)
is defined by
τ (t1v1 + · · ·+ tkvk) =
, . . . , tk −
1 · · · t
k 〈φi (v1, . . . , vk) , vα1...αk〉
α1...αk∈(Z2)k\{+···+}
Here Uk = {(ξ1, . . . , ξk) ∈ R
ξi = 0} is a Wk-module with an action given by
((β1, . . . , βk)⋊ π) · (ξ1, . . . , ξk) :=
ξπ−1(1), . . . , ξπ−1(k)
The subgroup (Z2)
k acts trivially on Uk. The action on Uk× (R2k)
is assumed to be the diagonal action.
The test map τ is well defined, continuous and Wk-equivariant.
Example 2.1. The test map τ : Xd,k → Uk × (R2k)
is in the case of k = 2 hyperplanes and j = 1
measure given by τ : Xd,2 → U2 ×R4 = U2 × ((V+− ⊕ V−+)⊕ V−−) and
τ (t1v1 + t2v2) =
, t2 −
t1〈φ (v1, v2) , v−+〉, t2〈φ (v1, v2) , v+−〉, t1t2〈φ (v1, v2) , v−−〉)
where
φ (v1, v2) =
∩Hα2v2 )−
µ(Rd)
α1α2∈(Z2)2
∈ R4.
2.3 The test space
The test spaces for the maps φ and τ are the origins of (R2k)
and Uk × (R2k)
, respectively. The
constructions that we perform in this section satisfy the usual hypotheses for the CS/TM scheme.
Proposition 2.2. (i) For a collection of mass distributions M = {µ1, . . . , µj} let φ : Yd,k → (R2k)
and τ : Xd,k → Uk × (R2k)
be the corresponding test maps. If
(0, . . . , 0) ∈ φ (Yd,k) or (0, . . . , 0) ∈ τ (Xd,k)
then there exists an arrangement of k hyperplanes H in Rd equiparting the collection M.
(ii) If there is no Wk-equivariant map with respect to the actions defined above,
Yd,k → (R2k)
\{(0, . . . , 0)}, or Yd,k → S
(R2k)
≈ Sj(2
k−1)−1, or
Xd,k → Uk × (R2k)
\{(0, . . . , 0)}, or Xd,k → S
Uk × (R2k)
≈ Sj(2
k−1)+k−2,
then the triple (d, j, k) is admissible.
(iii) Specifically, for k = 2, if there is no D8 ∼= W2 equivariant map, with the already defined actions,
Yd,2 → (R4)
\{(0, . . . , 0)}, or Yd,2 → S
≈ S3j−1, or
Xd,2 → U2 × (R4)
\{(0, . . . , 0)}, or S2d+1 ≈ Xd,2 → S
U2 × (R4)
≈ S3j ,
then the triple (d, j, 2) is admissible.
Remark 2.3. The action of Wk on the sphere S(U2 × (R4)
j) is fixed point free, but not free. For k = 2,
the action of the unique Z4 subgroup of W2 = D8 on the sphere S(U2 × (R4)
j) is fixed point free.
The necessary condition for the non-existence of an equivariant Wk-map
Xd,k → S(Uk × (R2k)
implied by the equivariant Kuratowski–Dugundji theorem [4, Theorem 1.3, page 25] is
dk + k − 1 > j(2k − 1) + k − 2 ⇐⇒ d ≥ 2
j . (7)
For k = 2 the condition (7) becomes
d ≥ ⌈ 3
j⌉. (8)
3 The Fadell–Husseini index theory
3.1 Equivariant cohomology
Let X be a G -space and X → EG×GX
→ BG the associated universal bundle, with X as a typical fibre.
EG is a contractible cellular space on which G acts freely, and BG := EG/G. The space EG ×G X =
(EG×X) /G is called the Borel construction of X with respect to the action of G. The equivariant
cohomology of X is the ordinary cohomology of the Borel construction EG×G X ,
H∗G(X) := H
∗(EG×G X).
The equivariant cohomology is a module over the ring H∗G(pt) = H
∗(BG). When X is a free G-space the
homotopy equivalence EG×G X ≃ X/G induces a natural isomorphism
H∗G(X)
∗(X/G).
The universal bundle X → EG ×G X
→ BG, for coefficients in the ring R, induces a Serre spectral
sequence converging to the graded group Gr(H∗G(X,R)) associated with H
G(X,R) appropriately filtered.
In this paper “ring” means commutative ring with a unit element. The E2-term is given by
p(BG,Hq(X,R)), (9)
where Hq(X,R) is a system of local coefficients. For a discrete group G, the E2-term of the spectral
sequence can be interpreted as the cohomology of the groupG with coefficients in the G-moduleH∗(X,R),
∼= Hp(G,Hq(X,R)). (10)
3.2 IndexG,R and Index
Let X be a G-space, R a ring and π∗X the ring homomorphism in cohomology
π∗X : H
∗(BG,R)→ H∗(EG×G X,R)
induced by the projection EG×G X → EG×G pt ≈ BG.
The Fadell–Husseini (ideal-valued) index of a G-space X is the kernel ideal of π∗X ,
IndexG,RX := kerπ
X ⊆ H
∗(BG,R).
The Serre spectral sequence (9) yields a representation of the homomorphism π∗X as the composition
H∗(BG,R)→ E
2 → E
3 → E
4 → · · · → E
∞ ⊆ H
∗(EG×G X,R).
The k-th Fadell–Husseini index is defined by
IndexkG,RX = ker
H∗(BG,R)→ E
, k ≥ 2,
Index1G,RX = {0}.
From the definitions the following properties of indexes can be derived.
Proposition 3.1. Let X, Y be G-spaces.
(1) IndexkG,RX ⊆ H
∗(BG,R) is an ideal, for every k ∈ N;
(2) Index1G,RX ⊆ Index
G,RX ⊆ Index
G,RX ⊆ · · · ⊆ IndexG,RX;
k∈N Index
G,RX = IndexG,RX.
Proposition 3.2. Let X and Y be G-spaces and f : X → Y a G-map. Then
IndexG,R(X) ⊇ IndexG,R(Y )
and for every k ∈ N
IndexkG,R(X) ⊇ Index
G,R(Y ).
Proof. Functoriality of all constructions implies that the following diagrams commute:
f ✲ Y EG×G X
f̂ ✲ EG×G Y
and consequently applying cohomology functor
H∗(EG×G X,R) ✛
H∗(EG×G Y,R)
H∗(BG,R)
πX = πY ◦ f̂ and π
X = f
∗ ◦ π∗Y . Thus kerπ
X ⊇ kerπ
Example 3.3. Sn is a Z2-space with the antipodal action. The action is free and therefore
EZ2 ×Z2 S
n ≃ Sn/Z2 ≈ RP
n ⇒ H∗
(Sn, R) ∼= H
∗(RPn, R).
1. R = F2: The cohomology ring H
∗(BZ2,F2) = H
∗(RP∞,F2) is the polynomial ring F2[t] where
deg(t) = 1. The Z2-index of S
n is the principal ideal generated by tn+1:
IndexZ2,F2S
n = Indexn+2
Z2,F2
Sn = 〈tn+1〉 ⊆ F2[t].
2. R = Z: The cohomology ring H∗(BZ2,Z) = H
∗(RP∞,Z) is the quotient polynomial ring Z[τ ]/〈2τ〉
where deg(τ) = 2. The Z2-index of S
n is the principal ideal
IndexZ2,ZS
n = Indexn+2
2 〉, for n odd,
2 〉, for n even.
Example 3.4. Let G be a finite group and H a subgroup of index 2. Then H ⊳G and G/H ∼= Z2. Let
V be the 1-dimensional real representation of G defined for v ∈ V by
g · v =
v, for g ∈ H,
−v, for g /∈ H.
There is a G-homeomorphism S(V ) ≈ Z2. Therefore by [21, last equation on page 34]:
EG×G S(V ) ≈ EG×G (G/H) ≈ (EG×G G) /H ≈ EG/H ≈ BH
IndexG,RS(V ) = ker
resGH : H
∗(G,R)→ H∗(H,R)
. (11)
3.3 The restriction map and the index
Let X be a G-space and K ⊆ G a subgroup. Then there is a commutative diagram of fibrations [12,
pages 179-180]:
EG×G X ✛
EG×K X
BG = EG/G
✛Bi EG/K = BK
induced by inclusion i : K ⊂ G. Here EG in the lower right corner is understood as a K-space and
consequently a model for EK. The map Bi is a map between classifying spaces induced by inclusion i.
Now with coefficients in the ring R we define
resGK := H
∗(f) : H∗(EG×G X,R)→ H
∗(EG×K X,R).
If G is a finite group, then the induced map on the cohomology of the classifying spaces
resGK = (Bi)
∗ : H∗(BG,R)→ H∗(BK,R)
coincides with the restriction homomorphism between group cohomologies
resGK : H
∗(G,R)→ H∗(K,R).
Proposition 3.5. Let X be a G-space, and K and L subgroups of G.
(A) The morphism of fibrations (12) provides the following commutative diagram in cohomology:
H∗(EG×G X,R)
resGK✲ H∗(EG×K X,R)
H∗(BG,R)
resGK ✲ H∗(BK,R)
(B) For every x ∈ H∗(BG,R) and y ∈ H∗(EG×G X,R),
resGK(x · y) = res
K(x) · res
K(y).
(C) L ⊂ K ⊂ G ⇒ resGL = res
L ◦ res
(D) The map of fibrations (12) induces a morphism of Serre spectral sequences
i : E
i (EG×G X,R)→ E
i (EK ×K X,R)
such that
(1) Γ∗,∗∞ = res
K : H
∗+∗(EG×G X,R)→ H
∗+∗(EG×K X,R),
(2) Γ
2 = res
K : H
∗(BG,R)→ H∗(BK,R).
(E) Let R and S be commutative rings and φ : R→ S a ring homomorphism. There are morphisms:
(1) in equivariant cohomology Φ∗ : H∗(EG×G X,R)→ H
∗(EG×G X,S),
(2) in group cohomology Φ∗ : H∗(G,R)→ H∗(G,S), and
(3) between Serre spectral sequences Φ
i : E
i (EG×G X,R)→ E
i (EG×G X,S),
induced by φ such that the following diagram commutes:
H∗(EG×G X,R) ✲ H
∗(EG×K X,R)
H∗(EG×G X,S) ✲
H∗(EG×K X,S)
H∗(BG,R)
✲ H∗(BK,R)
H∗(BG,S)
H∗(BK,S)
Remark 3.6. By a morphism of spectral sequences in properties (D) and (E) we mean that
i ◦ ∂i = ∂i ◦ Γ
i and Φ
i ◦ ∂i = ∂i ◦ Φ
These relations are applied in the situations where the right hand side is 6= 0 for a particular element x,
to imply that the left hand side Γ
i ◦ ∂i(x) or Φ
i ◦ ∂i(x) is also 6= 0. In particular, then ∂i(x) 6= 0.
Figure 1: Illustration of Proposition 3.5 (D) and (E)
Proposition 3.7. Let X be a G-space and K a subgroup of G. Let R and S be rings and φ : R → S a
ring homomorphism. Then
(1) resGK (IndexG,RX) ⊆ IndexK,RX,
(2) resGK
IndexrG,RX
⊆ IndexrK,RX for every r ∈ N,
(3) Φ∗(IndexG,RX) ⊆ IndexG,SX,
(4) Φ∗(IndexrG,RX) ⊆ Index
G,SX.
Proof. The assertions about the IndexG,R follow from diagrams (13) and (14). The commutative diagrams
E∗,0r (EG×G X,R)
Γ∗,0r✲ E∗,0r (EK ×K X,R)
H∗(BG,R)
resGK ✲ H∗(BK,R)
E∗,0r (EG×G X,R)
Φ∗,0r✲ E∗,0r (EG×G X,S)
H∗(BG,R)
Φ∗ ✲ H∗(BG,S)
imply the partial index assertions.
3.4 Basic calculations of the index
3.4.1 The index of a product
Let X be a G-space and Y an H-space. Then X × Y has the natural structure of a G×H-space. What
is the relation between the three indexes IndexG×H(X × Y ), IndexG(X), and IndexH(Y )? Using the
Künneth formula one can prove the following proposition [14, Corollary 3.2], [32, Proposition 2.7] when
the coefficient ring is a field.
Proposition 3.8. Let X be a G-space and Y an H-space and
H∗(BG, k) ∼= k[x1, . . . , xn], H
∗(BH, k) ∼= k[y1, . . . , ym]
the cohomology rings of the associated classifying spaces with coefficients in the field k. If
IndexG,kX = 〈f1, . . . , fi〉 and IndexH,k(Y ) = 〈g1, . . . , gj〉,
IndexG×H,kX = 〈f1, . . . , fi, g1, . . . , gj〉 ⊆ k[x1, . . . , xn, y1, . . . , ym].
The (Z2)
k-index of a product of spheres can be computed using this proposition and Example 3.3.
Corollary 3.9. Let Sn1 × · · · × Snk be a (Z2)
k-space with the product action. Then
Index(Z2)k,F2S
n1 × · · · × Snk = 〈tn1+11 , . . . , t
k 〉 ⊆ F2[t1, . . . , tk].
Unfortunately when the coefficient ring is not a field the claim of Proposition 3.8 does not hold.
Example 3.10. Let Sn × Sn be a (Z2)
2-space with the product action. From the previous corollary
Index(Z2)2,F2S
n × Sn = 〈tn+11 , t
2 〉 ⊆ F2[t1, t2] = H
∗((Z2)
2,F2). (15)
To determine the Z-index we proceed in two steps.
Cohomology ring H∗((Z2)
2,Z): Following [24, Section 4.1, page 508] the short exact sequence of
coefficients
→ F2 → 0 (16)
induces a long exact sequence in group cohomology [8, Proposition 6.1, page 71] which in this case reduces
to a sequence of short exact sequences for k > 0,
0→ Hk((Z2)
→ Hk((Z2)
2,F2)→ H
k+1((Z2)
2,Z)→ 0. (17)
Therefore, as in [24, Proposition 4.1, page 508],
H∗((Z2)
2,Z) ∼= (Z[τ1, τ2]⊗ Z[µ]) /I (18)
where deg τ1 = deg τ2 = 2, deg µ = 3 and the ideal I is generated by the relations
2τ1 = 2τ2 = 2µ = 0 and µ
2 = τ1τ2(τ1 + τ2).
The ring morphism c : Z → F2 in the coefficient exact sequence (16) induces a morphism in group
cohomology c∗ : H
∗((Z2)
2,Z)→ H∗((Z2)
2,F2) given by:
τ1 7−→ t
1, τ2 7−→ t
2, µ 7−→ t1t2(t1 + t2). (19)
The arguments used in the computation of the cohomology with integer coefficients come from the
Bockstein spectral sequence [7], [10, pages 104-110] associated with the exact couple
H∗((Z2)
2,Z) ✛
H∗((Z2)
H∗((Z2)
2,F2)
where deg(p) = deg(q) = 0 and deg(δ) = 1. The first differential d1 = q ◦ δ coincides with the first
Steenrod square Sq1 : H∗((Z2)
2,F2)→ H
∗+1((Z2)
2,F2) and therefore is given by
1 7→ 0, t1 7→ t
1, t2 7→ t
Consequently, t1t2 7→ t
1t2 + t1t
2. The spectral sequence stabilizes at the second step since the derived
couple is where F2 is in dimension 0.
0 ✛ 0
Index(Z2)2,ZS
n × Sn: The (Z2)
2-action on Sn × Sn, as a product of antipodal actions, is free and
therefore
E(Z2)
2 ×(Z2)2 (S
n × Sn) ≃ (Sn × Sn) /(Z2)
2 ≈ RPn × RPn.
Using equality (15), Proposition 3.5.E.3 on the coefficient morphism c : Z→ F2, the isomorphism
H∗(Z2)2(S
n × Sn,Z) ∼= H∗(RPn × RPn,Z)
and the existence of the (Z2)
2-inclusions
Sn−1 × Sn−1 ⊂ Sn × Sn ⊂ Sn+1 × Sn+1,
it can be concluded that
Index(Z2)2,ZS
n × Sn =
1 , τ
2 〉, for n odd
1 , τ
2 , τ
1 µ, τ
2 µ〉, for n even
⊆ H∗((Z2)
2,Z). (20)
3.4.2 The index of a sphere
We need to know how to compute the index of a sphere admitting an action of a finite group different
from the antipodal Z2-action. The following three propositions will be of some help [14, Proposition
3.13], [32, Proposition 2.9].
Proposition 3.11. Let G be a finite group and V an n-dimensional complex representation of G. Then
IndexG,ZS(V ) = 〈cn(VG)〉 ⊂ H
∗(G,Z)
where cn(VG) is the n-th Chern class of the bundle V → EG×G V → BG.
Proof. If the group G acts on H∗(S(V ),Z) trivially, then from the Serre spectral sequence of the sphere
bundle
S(V )→ EG×G S(V )→ BG
it follows that
IndexG,ZS(V ) = 〈e(VG)〉 ⊂ H
∗(G,Z),
where e(V ) is the Euler class of the bundle V → EG×G V → BG. Now V is a complex G-representation,
therefore the group G acts trivially on H∗(S(V ),Z). From [22, Exercise 3, page 261] it follows that
e(VG) = cn(VG)
and the statement is proved.
Proposition 3.12. Let U , V be two G-representations and let S(U), S(V ) be the associated G-spheres.
Let R be a ring and assume that H∗(S(U), R), H∗(S(V ), R) are trivial G-modules. If IndexG,R(S(U)) =
〈f〉 ⊆ H∗(BG,R) and IndexG,R(S(V )) = 〈g〉 ⊆ H
∗(BG,R), then
IndexG,RS(U ⊕ V ) = 〈f · g〉 ⊆ H
∗(BG,R).
Proposition 3.13. (A) Let V be the 1-dimensional (Z2)
k-representation with the associated ±1 vector
(α1, . . . , αk) ∈ (Z2)
k (as defined in Section 2). Then
Index(Z2)k,F2S(V ) = 〈ᾱ1t1 + · · ·+ ᾱktk〉 ⊆ F2[t1, . . . , tk],
where ᾱi = 0 if αi = 1, and ᾱi = 1 if αi = −1.
(B) Let U be an n-dimensional (Z2)
k-representation with a decomposition U ∼= V1 ⊕ · · · ⊕ Vn into 1-
dimensional (Z2)
k-representations V1, . . . , Vn. If (α1i, . . . , αki) ∈ (Z2)
k is the associated ±1 vector
of Vi, then
Index(Z2)k,F2S(U) =
(ᾱ1it1 + · · ·+ ᾱkitk)
⊆ F2[t1, . . . , tk].
Example 3.14. Let V−+, V+− and V−− be 1-dimensional real (Z2)
2-representations introduced in Section
2.2. Then by Proposition 3.13
Index(Z2)2,F2S(V−+) = 〈t1〉, Index(Z2)2,F2S(V+−) = 〈t2〉, Index(Z2)2,F2S(V−−) = 〈t1 + t2〉.
On the other hand, Example 3.4 and the restriction diagram (42) imply that
Index(Z2)2,ZS(V−+) = 〈τ1, µ〉, Index(Z2)2,ZS(V+−) = 〈τ2, µ〉, Index(Z2)2,ZS(V−−) = 〈τ1 + τ2, µ〉.
4 The cohomology of D8 and the restriction diagram
The dihedral group W2 = D8 = (Z2)
2 ⋊ Z2 = (〈ε1〉 × 〈ε2〉)⋊ 〈σ〉 can be presented by
D8 = 〈ε1, σ | ε
1 = σ
2 = (ε1σ)
= 1〉.
Then 〈ε1σ〉 ∼= Z4 and ε2 = σε1σ.
4.1 The poset of subgroups of D8
The poset Sub(G) denotes the collection of all nontrivial subgroups of a given group G ordered by
inclusion. The poset Sub(G) can be interpreted as a small category G in the usual way:
• Ob(G) = Sub(G),
• for every two objects H and K, subgroups of G, there is a unique morphism fH,K : H → K if
H ⊇ K, and no morphism if H + K, i.e.
Mor(H,K) =
{fH,K } , H ⊇ K,
∅ , H + K.
The Hasse diagram of the poset Sub(D8) is presented in the following diagram.
〈ε1, ε2〉
Z2 × Z2
〈ε1σ〉
〈ε1ε2, σ〉
Z2 × Z2
〈ε1ε2〉
〈ε1ε2σ〉
4.2 The cohomology ring H∗(D8,F2)
The dihedral group D8 is an example of a wreath product. Therefore the associated classifying space
can, as in [1, page 117], be written explicitly as
BD8 = B(Z2)
2 ×Z2 EZ2 ≈ (B(Z2)
2)×Z2 EZ2,
where Z2 = 〈σ〉 acts on (BZ2)
2 by interchanging coordinates. Presented in this way BD8 is the Borel
construction of the Z2-space (BZ2)
2. Thus BD8 fits into a fibration
B(Z2)
2 → (B(Z2)
2)×Z2 EZ2 → BZ2. (21)
There is an associated Serre spectral sequence with E2-term
Hp(BZ2,H
B(Z2)
Hp(Z2, H
Hp+q(BD8,F2)
Hp+q(D8,F2)
which converges to the cohomology of the group D8 with F2-coefficients. This spectral sequence is also
the Lyndon-Hochschild-Serre (LHS) spectral sequence [1, Section IV.1, page 116] associated with the
group extension sequence:
1→ (Z2)
2 → D8 → D8/(Z2)
2 → 1.
In [1, Theorem 1.7, page 117] it is proved that the spectral sequence (22) collapses at the E2-term.
Therefore, to compute the cohomology of D8 we only need to read the E2-term.
Lemma 4.1.
(i) H∗
∼=ring F2[a, a+ b], where deg(a) = deg(a + b) = 1 and the Z2-action induced by σ is
given by σ · a = a+ b.
(ii) H∗
)Z2 ∼=ring F2[b, a(a+ b)].
(iii) Hi
∼=Z2-module F2[Z2]
si,1 ⊕ F
2 , where si,1 ≥ 0, si,2 ≥ 0 and F2[Z2] denotes a free
Z2-module and F2 a trivial one.
(iv) E
2 = H
∗(Z2, H
) ∼=ring H
∗(Z2,F2)
⊕si,2 ⊕F
2 , where F
2 denotes a ring concentrated
in dimension 0.
Proof. (i) The statement follows from the observation that B(Z2)
2 ≈ (B(Z2))
, and consequently
∼=ring H
∗ (Z2,F2)⊗H
∗ (Z2,F2) ∼=ring F2[a]⊗ F2[a+ b].
The Z2-action interchanges copies on the left hand side. Generators on the right hand side are chosen
such that the Z2-action coming from the isomorphism swaps a and a+ b.
(ii) With the induced Z2-action b = a+(a+ b) and a(a+ b) are invariant polynomials. They generate
the ring of all invariant polynomials.
(iii) The cohomology Hi
is a Z2-module and therefore a direct sum of irreducible Z2-
modules. There are only two irreducible Z2-modules over F2: the free one F2[Z2] and the trivial one F2.
(iv) The isomorphism follows from (iii) and the following two properties of group cohomology [20,
Exercise 2.2, page 190] and [8, Corollary 6.6, page 73]. Let M and N be G-modules of a finite group G.
(a) H∗(G,M ⊕N) ∼= H∗(G,M)⊕H∗(G,N)
(b) M is a free G-module ⇒ H∗(G,M) = H0(G,M) ∼= MG.
Applied in our case, this yields
2 =ring H
∗(Z2, H
∼=ring H
∗(Z2,F2[Z2]
si,1 ⊕ F
∼=ring H
∗(Z2,F2[Z2])
⊕si,1 ⊕H∗(Z2,F2)
⊕si,2
∼=ring H
0(Z2,F2[Z2])
⊕si,1 ⊕H∗(Z2,F2)
⊕si,2
∼=ring (F2[Z2]
Z2)⊕si,1 ⊕H∗(Z2,F2)
⊕si,2
∼=ring F2
⊕si,1 ⊕H∗(Z2,F2)
⊕si,2
Let the cohomology of the base space of the fibration (21) be denoted by
H∗(Z2,F2) = F2[x].
The E2-term (22) can be pictured as in Figure 2.
The cohomology of D8 can be read from the picture. If we denote
y := b, w := a(a+ b) (23)
and keep x as we introduced above, then
H∗(D8,F2) = F2[x, y, w]/〈xy〉.
Also, the restriction homomorphism
resD8H1 : H
∗(D8,F2) = F2[x, y, w]/〈xy〉 → H
∗(H1,F2) = F2[a, a+ b] (24)
can be read off since it is induced by the inclusion of the fibre in the fibration (21). On generators,
resD8H1(x) = 0, res
(y) = b, resD8H1(w) = a(a+ b). (25)
b4,b2aÝa + bÞ
a2Ýa + bÞ2 å 1
a2Ýa + bÞ2 å x a2Ýa + bÞ2 å x a2Ýa + bÞ2 å x a2Ýa + bÞ2 å x
3 b3,baÝa + bÞ 0 0 0 0
aÝa + bÞ å 1
aÝa + bÞ å x aÝa + bÞ å x2 aÝa + bÞ å x3 aÝa + bÞ å x4
1 b 0 0 0 0
0 1 x x2 x3 x4
0 1 2 3 4
Figure 2: E2-term
4.3 The cohomology diagram of subgroups with coefficients in F2
Let G be a finite group and R an arbitrary ring. Then the diagram Res(R) : G→ Ring (covariant functor)
defined by
Ob(G) ∋ H 7−→ H∗(H,R)
( H ⊇ K) 7−→
resHK : H
∗(H,R)→ H∗(K,R)
is the cohomology diagram of subgroups of G with coefficients in the ring R. In this section we assume
that R = F2.
4.3.1 The Z2 × Z2-diagram
The cohomology of any elementary abelian 2-group Z2 × Z2 is a polynomial ring F2 [x, y], deg(x) =
deg(y) = 1. The restrictions to the three subgroups of order 2 are given by all possible projections
F2 [x, y]→F2 [t], deg(t) = 1:
(x 7→ t, y 7→ 0) or (x 7→ 0, y 7→ t) or (x 7→ t, y 7→ t) .
Thus the cohomology diagram of the subgroups of Z2 × Z2 is
Z2 × Z2
F2 [x, y]
F2[t1]
F2[t2]
x 7→ t2 y 7→ 0
F2[t3]
4.3.2 The D8-diagram
For the dihedral group D8, from [9] and (24), the two top levels of the diagram can be presented by:
F2 [x, y;w]/〈xy 〉
deg : 1, 1, 2
F2 [a, b]
deg : 1, 1
F2 [e, u]/〈e
deg : 1, 2
x, y 7→ e w 7→ u
F2 [c, d]
deg : 1, 1
Let H∗(Ki,F2) = F2[ti], deg(ti) = 1. From [1, Corollary II.5.7, page 69] the restriction
resH2K3 :
H∗(H2,F2) = F2[e, u] /〈e
−→ (H∗(K3,F2) = F2[t3])
is given by e 7→ 0, u 7→ t23. Thus, the restriction res
is given by x 7→ 0, y 7→ 0, w 7→ t23. Using diagrams
(26), (27) with the property (C) from Proposition 3.5 we almost completely reveal the cohomology
diagram of subgroups of D8. The equalities
resD8K3 = res
◦ resD8H2 = res
◦ resD8H1 = res
◦ resD8H3
imply that
• resH1K3 : (H
∗(H1,F2) = F2[a, b]) −→ (H
∗(K3,F2) = F2[t3]) is given by a 7→ t3, b 7→ 0,
• resH3K3 : (H
∗(H3,F2) = F2[c, d]) −→ (H
∗(K3,F2) = F2[t3]) is given by c 7→ t3, d 7→ 0.
F2 [a, b]
deg : 1, 1
F2 [e, u]/〈e
deg : 1, 2
F2 [c, d]
deg : 1, 1
F2[t3]
deg : 1
u 7→ t23 e 7→ 0❄
The cohomology diagram (26) of subgroups of Z2 × Z2 and the part (28) of the D8 diagram imply that
• resH1K1 : F2[a, b] −→ F2[t1] and res
: F2[a, b] −→ F2[t2] are given by
(a 7→ t1, b 7→ t1 and a 7→ 0, b 7→ t2) or (a 7→ 0, b 7→ t1 and a 7→ t2, b 7→ t2) ,
• resH3K4 : F2[c, d] −→ F2[t4] and res
: F2[a, b] −→ F2[t5] are given by
(c 7→ t4, d 7→ t4 and c 7→ 0, d 7→ t5) or (c 7→ 0, d 7→ t4 and c 7→ t5, d 7→ t5) .
Proposition 4.2. For all i 6= 3, resD8Ki (w) = 0, while res
(w) 6= 0.
Proof. The result follows from the diagram (27) in the following way:
(a) For i ∈ {1, 2}:
resD8Ki (w) = res
◦ resD8H1(w) = res
(a(a+ b)) = 0
since either a 7→ ti, b 7→ ti or a 7→ 0, b 7→ ti.
(b) For i ∈ {4, 5}:
resD8Ki (w) = res
◦ resD8H3(w) = res
(c(c+ d)) = 0
since either c 7→ ti, d 7→ ti or c 7→ 0, d 7→ ti.
Corollary 4.3. The cohomology of the dihedral group D8 is
H∗(D8,F2) = F2[x, y, w]/〈xy〉
where
(a) x ∈ H1(D8,F2) and res
(x) = 0,
(b) y ∈ H1(D8,F2) and res
(y) = 0,
(c) w ∈ H1(D8,F2) and res
(w) = resD8K2(w) = res
(w) = resD8K5(w) = 0 and res
(w) 6= 0.
Assumption Without lose of generality we can assume that
resH1K1(a) = t1, res
(b) = t1, res
(a) = 0, resH1K2(b) = t2. (29)
4.4 The cohomology ring H∗(D8,Z)
In this section we present the cohomology H∗(D8,Z) based on:
A. Evens’ approach [13, Section 5, pages 191-192], where the concrete generators in H∗(D8,Z) are
identified using the Chern classes of appropriate irreducible complex D8-representations. We also
consider LHS spectral sequences associated with following two extensions
1→ H1 → D8 → D8/(Z2)
2 → 1 and 1→ H2 → D8 → D8/Z4 → 1. (30)
Unfortunately, the ring structure on E∞ -terms of these LHS spectral sequences does not coincide
with the ring structure on H∗(D8,Z).
B. The Bockstein spectral sequence of the exact couple
H∗(D8,Z)
×2 ✲ H∗(D8,Z)
H∗(D8,F2)
where d1 = c ◦ δ = Sq
1 : H∗(D8,F2) → H
∗+1(D8,F2) is given by d1 (x) = x
2, d1 (y) = y
2 and
d1 (w) = (x + y)w [1, Theorem 2.7. page 127]. This approach allows determination of the ring
structure on H∗(D8,Z).
4.4.1 Evens’ view
Let V C+− ⊕ V
−+ = C⊕ C, V
−− = C and U
2 = C be the complex D8-representations given by
A. For (u, v) ∈ V C+− ⊕ V
ε1 · (u, v) = (u,−v), ε2 · (u, v) = (−u, v), σ · (u, v) = (v, u).
B. For u ∈ V C−−:
ε1 · u = −u, ε2 · u = −u, σ · u = u.
C. For u ∈ UC2 :
ε1 · u = u, ε2 · u = u, σ · u = −u.
There are isomorphisms of real D8-representations
V C+− ⊕ V
∼= (V+− ⊕ V−+)
, V C−−
∼= (V−−)
, UC2 = (U2)
Let χ1, ξ ∈ H
∗(D8,Z) be 1-dimensional complex D8-representations given by character (here we assume
the identification c1 : Hom(G,U (1))→ H
2(G,Z), [3, page 286]):
χ1(ε1) = 1, χ1(ε2) = 1, χ1(σ) = −1,
ξ(ε1) = −1, ξ(ε2) = −1, ξ(σ) = −1.
Then χ1 = U
2 , ξ = U
2 ⊗ V
−− and consequently
2 ) = χ1, and c1(U
2 ) + c1(V
−−) = ξ. (31)
The cohomology H∗(D8,Z) is given in [13, pages 191-192] by
H∗(D8,Z) = Z[ξ, χ1, ζ, χ] (32)
where
deg ξ = degχ1 = 2, deg ζ = 3, degχ = 4
2ξ = 2χ1 = 2ζ = 4χ = 0, χ
1 = ξ · χ1, ζ
2 = ξ · χ. (33)
There are four 1-dimensional irreducible complex representations of D8:
1, ξ = UC2 ⊗ V
−−, χ1 = U
2 , ξ ⊗ χ1 = V
and one 2-dimensional complex representation which is denoted by ρ in [13, pages 191-192]:
ρ = V C+− ⊕ V
It is computed in [13, pages 191-192] that
c(V C+− ⊕ V
−+) = 1 + ξ + χ and c2(V
+− ⊕ V
−+) = χ. (34)
The relations (31) and (34) along with Proposition 3.11 imply the following statement.
Proposition 4.4. IndexD8,ZS(V
−−) = 〈ξ+χ1〉, IndexD8,ZS(U
2 ) = 〈χ1〉, IndexD8,ZS(V
−+) = 〈χ〉.
Before proceeding to the Bockstein spectral sequence approach we give descriptions of the E2-terms of
two LHS spectral sequences. Even though it is not an easy consequence, it can be proved that both
spectral sequences stabilize and that E2 = E∞.
LHS spectral sequences of the extension 1 → H1 → D8 → D8/H1 → 1. The LHS spectral
sequence of this extension (22) allows computation of the cohomology ringH∗(D8,F2) with F2 coefficients.
If we now consider Z coefficients, then the E2-term has the form
2 = H
p(D8/H1, H
q (H1,Z)) ∼= H
p(Z2, H
). (35)
The spectral sequence converges to the graded group Gr (Hp+q(D8,Z)) associated with H
p+q(D8,Z)
appropriately filtered. To present the E2-term we choose generators of H
∗ (H1,Z) consistent with the
choices made in Lemma 4.1. Let c : Z → F2 be reduction mod 2 and c∗ : H
∗(D8,Z) → H
∗(D8,F2) the
induced map in cohomology. Consider the following presentation of the H1 cohomology ring:
H∗ (H1,Z) = Z[α, α+ β]⊗ Z[µ] (36)
where
A. deg(α) = deg(β) = 2, deg(µ) = 3;
B. 2α = 2β = 2µ = 0 and µ2 = αβ(α + β);
C. σ action on H∗ (H1,Z) is given by σ · α = α+ β and σ · µ = µ;
D. c∗(α) = a
2, c∗ (β) = b
2, c∗ (µ) = ab(a+ b).
å3 0 0
1 0 0 0 0 0
0 1 x2
0 1 2 3 4
0 0 0 0
m xåm åmm x2
a(a+b), b a(a+b)åxå0 0 åx
2a(a+b)2
Figure 3: E2-term of extension 1 → H1 → D8 → D8/H1 → 1.
Now the E2-term (Figure 3) is given by
∼= Hp(Z2, H
Hp(Z2,Z), q = 0
0, q = 1
Hp(Z2,F2[Z2]), q = 2
Hp(Z2,F2), q = 3
. . . , q > 3.
The morphism of LHS spectral sequences of the extension 1→ H1 → D8 → D8/H1 → 1 induced by the
mod 2 reduction c : Z → F2 (Proposition 3.5 E.3) gives a proof that E2 = E∞ for Z coefficients. The
ring structures on E∞ and H
∗(D8,Z) do not coincide. Moreover there is no element in E∞ of exponent
4. One thing is clear: the element µ in the E2 = E∞-term coincides with the element ζ in the Evens’
presentation (32) of H∗(D8,Z).
LHS spectral sequences of the extension 1 → H2 → D8 → D8/H2 → 1. The E2-term has the
form:
2 = H
p(D8/H2, H
q (H2,Z)) ∼= H
p(Z2, H
q (Z4,Z)) ∼=
Hp(Z2,Z), q = 0
0, q odd
Hp(Z2,Z4), q even and 4 ∤ q
Hp(Z2,Z4), q > 0 even and 4|q,
where Z4 ∼= Z4 is a non-trivial Z2-module. Using [8, Example 2, pages 58-59] the E2-term has the shape
given in the Figure 4. This diagram provides just two hints: there might be elements of exponent 4
6 ¨2 ¨2 ¨2 ¨2 ¨2 ¨2 ¨2
5 0 0 0 0 0 0 0
4 ¨4 ¨2 ¨2 ¨2 ¨2 ¨2 ¨2
3 0 0 0 0 0 0 0
2 ¨2 ¨2 Q ¨2 ¨2 ¨2 ¨2 ¨2
1 0 0 0 0 0 0 0
0 ¨ 0 ¨2 0 ¨2 0 ¨2
0 1 2 3 4 5 6
Figure 4: E2-term of extension 1 → H2 → D8 → D8/H2 → 1.
in the cohomology H∗(D8,Z) and definitely there is only one element ζ of degree 3 from the Evens’
presentation.
Conclusion. The LHS spectral sequences of different extensions gives an incomplete picture of the
cohomology ring with integer coefficients, H∗(D8,Z). Therefore, for the purposes of the computations
with Z coefficients we use the Bockstein spectral sequence utilizing results obtained from the LHS spectral
sequence with F2 coefficients. Presentations of these two spectral sequences will be used in the description
of the restriction diagram in Section 4.5.
4.4.2 The Bockstein spectral sequence view
Let G be a finite group. The exact sequence 0 → Z
→ Z → F2 → 0 induces a long exact sequence in
group cohomology, or an exact couple
H∗(G,Z)
×2 ✲ H∗(G,Z)
H∗(G,F2).
The spectral sequence of this exact couple is the Bockstein spectral sequence. It converges to
(H∗(G,Z)/torsion)⊗ F2
which in the case of a finite group G is just F2 in dimension 0. Here “torsion” means Z-torsion. The first
differential d1 = c ◦ δ is the Bockstein homomorphism and in this case coincides with the first Steenrod
square Sq1 : H∗(G,F2)→ H
∗+1(G,F2).
Let H be a subgroup of G. The restriction resGH commutes with the maps in the exact couples associated
to the groups G and H and therefore induces a morphism of Bockstein spectral sequences [10, page 109
before 5.7.6].
Consider two Bockstein spectral sequences associated with D8 and its subgroup H2 ∼= Z4.
A. Group D8. The exact couple is
H∗(D8,Z)
×2 ✲ H∗(D8,Z)
H∗(D8,F2).
and d1 = c ◦ δ = Sq
1 is given by d1 (x) = x
2, d1 (y) = y
2 and d1 (w) = (x + y)w, [1, Theorem 2.7.
page 127]. The derived couple is Then by [10, Remark 5.7.4, page 108] there are elements X ,Y ∈
2 ·H∗(D8,Z)
×2 ✲ 2 ·H∗(D8,Z)
〈x2, y2, xw, yw,w2〉/〈x2, y2, xw + yw〉.
H2(D8,Z), M ∈ H
3(D8,Z) of exponent 2 such that c∗(X ) = x
2, c∗(Y) = y
2, c∗(M) = (x + y)w
and XY = 0.
B. Group Z4. The exact couple is
H∗(Z4,Z)
×2 ✲ H∗(Z4,Z)
H∗(Z4,F2).
Since H∗(Z4,Z) = Z[U ]/〈4U〉, degU = 2 and H
∗(Z4,F2) = F2 [e, u]/〈e
〉, deg e = 1, deg u = 2, the
unrolling of the exact couple to a long exact sequence [8, Proposition 6.1, page 71]
H0(Z4,Z)
Z , 1
×2✲ H
0(Z4,Z)
Z , 1
0(Z4,F2)
F2 , 1
1(Z4,Z)
×2✲ H
1(Z4,Z)
1(Z4,F2)
F2 , e
2(Z4,Z)
Z4 , U
×2✲ H
2(Z4,Z)
Z4 , U
2(Z4,F2)
F2 , u
3(Z4,Z)
×2✲ H
3(Z4,Z)
3(Z4,F2)
F2 , eu
4(Z4,Z)
. . .
allows us to show that for j ≥ 0 :
δ(ui) = 0 and δ(eui) = 2U i+1.
Thus d1 = 0 and the derived couple is
2 ·H∗(Z4,Z)
×2 ✲ 2 ·H∗(Z4,Z)
H∗(Z4,F2).
Moreover, by definition of the differential of a derived couple we have that
i) = 0 and d2(eu
i) = ui+1.
The restriction map resD8H2 : H
∗(D8,F2) → H
∗(H2,F2) is determined by the restriction diagram (27).
Therefore, the morphism between spectral sequences induced by the restriction resD8H2 implies that:
resD8H2 (d2[xw]) = d2
resD8H2 [xw]
= d2(eu) = u
and consequently
d2[xw] = [w
Here [ · ] denotes the class in the quotient 〈x2, y2, xw, yw,w2〉/〈x2, y2, xw + yw〉. Thus, by [10, Remark
5.7.4, page 108] there is an element W ∈ H4(D8,Z) of exponent 4 such that c∗(W) = w
2 and M2 =
W(X + Y). The second derived couple of (37) stabilizes. Thus the cohomology ring H∗(D8,Z) and the
map c∗ : H
∗(D8,Z) −→ H
∗(D8,F2) are described.
Theorem 4.5. The cohomology ring H∗(D8,Z) can be presented by
H∗(D8,Z) = Z[X ,Y,M,W ]/I
where
degX = degY = 2, degM = 3, degW = 4
and the ideal I is generated by the equations
2X = 2Y = 2M = 4W = 0, XY = 0,M2 =W(X + Y). (39)
The map c∗ : H
∗(D8,Z) −→ H
∗(D8,F2), induced by the reduction of coefficients Z→ F2, is given by
X 7→x2 , Y 7→y2 , M 7→w(x + y) , W 7→w2. (40)
Remark 4.6. The correspondence between the Evens’ and Bockstein spectral sequence view is given by
X ↔ χ1 , Y ↔ ξ + χ1 , M↔ ζ , W ↔ χ (41)
4.5 The D8-diagram with coefficients in Z
Let G be a finite group and R and S rings. A ring homomorphism φ : R → S induces a morphism of
diagrams (natural transformation of covariant functors) Φ : Res(R) →Res(S). The morphism Φ on each
object H ∈ Ob(G) is defined by the coefficient reduction Φ(H) : H∗(H,R) → H∗(H,S) induced by φ.
Particularly in this section, as a tool for the reconstruction of the diagram Res(Z), we use the diagram
morphism C : Res(Z) → Res(F2) induced by the coefficient reduction homomorphism c : Z→ F2.
4.5.1 The Z2 × Z2-diagram
The cohomology restriction diagram Res(F2) of the elementary abelian 2-group Z2 × Z2 is given in the
diagram (26). Using the presentation of cohomology H∗(Z2 × Z2,Z) and the homomorphism H
∗(Z2 ×
Z2,Z)→ H
∗(Z2 × Z2,F2) given in Example 3.10 we can reconstruct the restriction diagram Res(Z):
Z2 × Z2 Z[τ1, τ2]⊗ Z[µ]
deg τ1 = deg τ2 = 2, deg µ = 3
2τ1 = 2τ2 = 2µ = 0,
µ2 = τ1τ2(τ1 + τ2)
3 , τ
Z2 Z[θ1]
deg θ1 = 2
2θ1 = 0
Z2 Z[θ2]
deg θ2 = 2
2θ2 = 0
τ1 7→θ2,
τ2 7→ 0,
µ 7→ 0
Z2 Z[θ3]
deg θ3 = 2
2θ3 = 0
4.5.2 The D8-diagram
In a similar fashion, using:
• the D8 restriction diagram (27) and (28) with F2 coefficients,
• the Z2 × Z2 restriction diagrams (42) with Z coefficients,
• the presentation of cohomology H∗(H1,Z) given in (36),
• the Bockstein presentation of H∗(D8,Z) given in Theorem 4.5,
• a glance at the restriction maps resD8H1 and res
obtained from the E2 = E∞ terms of the LHS
spectral sequences Figure 3 and Figure 4, and
• the homomorphism c∗ : H
∗(D8,Z)→ H
∗(D8,F2) described in (40),
we can reconstruct the restriction diagram of D8 with Z coefficients.
D8 Z[X ,Y,M,W ]
deg : 2, 2, 3, 4
2X = 2Y = 2M = 4W = 0,
XY = 0,M2 =W(X + Y)
H1 Z[α, α + β, µ]
deg : 2, 2, 3,
2α = 2β = 2µ = 0,
µ2 = αβ(α + β)
H2 Z[U ]
deg : 2
4U = 0,
Y 7→ 2U
W7→ U2
X 7→2U
H3 Z[γ, γ + δ, η]
deg : 2, 2, 3,
2γ = 2δ = 2η = 0,
η2 = γδ(γ + δ)
K3 Z[θ3] deg θ3 = 2
U 7→θ3
Now the determination of the diagram morphism C :Res(Z) →Res(F2) induced by the coefficient reduction
homomorphism c : Z→ F2 is just a routine exercise.
5 IndexD8,F2S(R
In this section we show the following equality:
IndexD8,F2S(R
4 ) = Index
D8,F2
4 ) = 〈w
jyj〉.
The D8-representation R
4 can be decomposed into a sum of irreducibles in the following way
R4 = (V−+ ⊕ V+−)⊕ V−− ⇒ R
4 = (V−+ ⊕ V+−)
where V−+ ⊕ V+− is a 2-dimensional irreducible D8-representation. Since in this section F2 coefficients
are assumed, Proposition 3.12 implies that computing the indexes of the spheres S(V−+ ⊕ V+−) and
S(V−−) suffices. The strategy employed uses Proposition 3.7 and the following particular facts.
A. Let X = S(T ) for some D8-representation T . Then the E2-term of the Serre spectral sequence
associated to ED8 ×D8 X is
2 = H
p(D8,F2)⊗H
q(X,F2). (44)
The local coefficients are trivial since X is a sphere and the coefficients are F2. Since only ∂dimT,F2
may be 6= 0, from the multiplicative property of the spectral sequence it follows that
IndexD8,F2X = 〈∂
0,dimV −1
dimV,F2
(1⊗ l)〉
where l ∈ HdimV −1(X,F2) is the generator. Therefore, IndexD8,F2(X) = Index
dimV+1
D8,F2
B. For any subgroup H of D8, with some abuse of notation,
dimV,0
dimV ◦ ∂
0,dimV−1
dimV,F2
(1⊗ l) = ∂
0,dimV −1
dimV,F2
0,dimV −1
dimV (1⊗ l), (45)
where Γ denotes the restriction morphism of Serre spectral sequences introduced in Proposition
3.5(D). Therefore, for every subgroup H of D8 we get
IndexD8,F2X = 〈a〉, IndexH,F2X = 〈aH〉 =⇒ res
K(a) = aH .
In particular, if aH 6= 0 then a 6= 0.
Our computation of IndexD8,F2X for X = S(V−+ ⊕ V+−) and X = S(V−−) has two steps:
• compute IndexH,F2X = 〈aH〉 for all proper subgroups H of D8,
• search for an element a ∈ H∗(D8,F2) such that for every computed aH
resGK(a) = aH .
5.1 IndexD8,F2S(V−+ ⊕ V+−) = 〈w〉
Proposition 3.13 and the properties of the action of D8 on V−+⊕V+− provide the following information:
IndexH1,F2S(V−+ ⊕ V+−) =
〈a(a+ b)〉 or
〈b(a+ b)〉 or
〈ab〉.
Since initially we do not know which of the possible generators a, b, a + b of F2[a, b] correspond to the
generators ε1, ε2, ε1ε2, we have to take all three possibilities into account. Similarly:
IndexH3,F2S(V−+ ⊕ V+−) =
〈c(c+ d)〉 or
〈d(c+ d)〉 or
〈cd〉.
Furthermore,
ε1 acts trivially on V+− ⇒ IndexK1,F2S(V−+ ⊕ V+−) = 0
ε2 acts trivially on V−+ ⇒ IndexK2,F2S(V−+ ⊕ V+−) = 0
σ acts trivially on {(x, x) ∈ V−+ ⊕ V+−} ⇒ IndexK4,F2S(V−+ ⊕ V+−) = 0
ε1ε2σ acts trivially on {(x,−x) ∈ V−+ ⊕ V+−} ⇒ IndexK5,F2S(V−+ ⊕ V+−) = 0.
The only nonzero element of H2(D8,F2) satisfying all requirements of commutativity with restrictions
is w. Hence,
IndexD8,F2S(V−+ ⊕ V+−) = 〈w〉. (46)
Remark 5.1. The side information coming from this computation is that generators ε1 and ε2 of the
group H1 correspond to generators a and a+ b in the cohomology ring H
∗(H1,F2). This correspondence
suggested the choice of generators in Lemma 4.1(i).
5.2 IndexD8,F2S(V−−) = 〈y〉
Again, V−− is a concrete D8-representation, and from Proposition 3.13:
IndexH1,F2S(V−−) =
〈a+ b〉, or
〈a+ (a+ b)〉, or
〈b+ (a+ b)〉.
Again, we allow all three possibilities since we do not know the correspondence between generators of H1
and the chosen generators of H∗(Hq,F2). Furthermore, since K1 and K2 act nontrivially on V−−,
IndexK1,F2S(V−−) = 〈t1〉, IndexK2,F2S(V−−) = 〈t2〉 .
On the other hand, H3 acts trivially on S(V−−) and so
IndexH3,F2S(V−−) = 0.
By commutativity of the restriction diagram, or since the groups K3, K4 and K5 act trivially on V(1,1),
it follows that
IndexK3,F2S(V−−) = IndexK4,F2S(V−−) = IndexK5,F2S(V−−) = 0.
The only element satisfying the commutativity requirements is y ∈ H1(D8,F2), so
IndexD8,F2S(V−−) = 〈y〉. (47)
Remark 5.2. From the previous remark the fact that IndexH1,F2S(V−−) = 〈b〉 = 〈a + (a + b)〉 follows
directly. Alternatively, equation (47) is a consequence of (11) and (27).
5.3 IndexD8,F2S(R
4 ) = 〈y
From Proposition 3.12 we get that
IndexD8,F2S(R
4 ) = IndexD8,F2S((V−+ ⊕ V+−)
⊕j ⊕ V
−−) = 〈y
jwj〉.
Remark 5.3. In the same way we can compute that
IndexD8,F2(U2) = 〈x〉. (48)
Therefore IndexD8,F2(U2 ⊕ R
4 ) = 0. This means that on the join CS/TM scheme the Fadell–Husseini
index theory with F2 coefficients yields no obstruction to the existence of the equivariant map in question.
6 IndexD8,ZS(R
In this section we show that
IndexD8,ZS(R
4 ) = Index
4 ) =
2 〉, for j even
2 M,Y
2 〉, for j odd.
6.1 The case when j is even
We give two proofs of the equation (49) in the case when j is even.
Method 1: According to definition of the complex D8 -representations V
+− ⊕ V
−+ and V
−−, in Section
4.4.1, we have an isomorphism of real D8-representations
4 = (V−+ ⊕ V+−)
V C+− ⊕ V
V C−−
Thus by Propositions 3.11 and 3.12, properties of Chern classes [3, (5) page 286] and equations (31) and
(34) we have that
IndexD8,ZS(R
4 ) = 〈c 3j
V C+− ⊕ V
V C−−
〉 = 〈c2
V C+− ⊕ V
2 · c1
V C−−
2 (ξ + χ1)
The correspondence between Evens’ and Bockstein spectral sequence views implies the statement.
Method 2: The group D8 acts trivially on the cohomologyH
∗(S(R
4 ),Z). Then the E2-term of the Serre
spectral sequence associated to ED8 ×D8 S(R
4 ) is a tensor product
2 = H
p(D8,Z)⊗H
q(S(R
4 ),Z).
Since only ∂3j,Z may be 6= 0, the multiplicative property of the spectral sequence implies that
IndexD8,ZS(R
4 ) = Index
dimV +1
4 ) = 〈∂
0,3j−1
3j,Z (1 ⊗ l)〉
where l ∈ H3j−1(S(R
4 ),Z) is a generator. The coefficient reduction morphism c : Z → F2 induces a
morphism of Serre spectral sequences (Proposition 3.5. E. 3) associated with the Borel construction of
the sphere S(R
4 ). Thus,
0,3j−1
3j,Z (1 ⊗ l)
0,3j−1
3j,F2
(s∗(1 ⊗ l)) ∈ H
3j(D8,F2)
and according to the result of the previous section
0,3j−1
3j,Z (1⊗ l)
= yjwj .
Now, from the description of the map c∗ : H
∗(D8,Z) −→ H
∗(D8,F2) in (40) follows the statement for j
even.
6.2 The case when j is odd
The group D8 acts nontrivially on the cohomology H
∗(S(R
4 ),Z). Precisely, the D8-module Z =
H3j−1(S(R
4 ),Z) is a nontrivial D8-module and for z ∈ Z:
ε1 · z = z, ε2 · z = z, σ · z = −z.
Then the E2-term of the Serre spectral sequence associated to ED8 ×D8 S(R
4 ) is not a tensor product
2 = H
p(D8, H
q(S(R
4 ),Z)) =
Hp(D8,Z) , q = 0
Hp(D8,Z) , q = 3j − 1
0 , q 6= 0, 3j − 1.
To compute the index in this case we have to study the H∗(D8,Z)-module structure of H
∗(D8,Z).
Since the use of LHS-spectral sequence, as in the case of field coefficients (Proposition 7.4), cannot be of
significant help we apply the Bockstein spectral sequence associated with the following exact sequence of
D8-modules:
→ Z → F2 → 0. (51)
Proposition 6.1.
(A) 2 ·H∗(D8,Z) = 0
(B) H∗(D8,Z) is generated as a H
∗(D8,Z)-module by three elements ρ1, ρ2, ρ3 of degree 1, 2, 3 such
ρ1 · Y = 0, ρ2 · X = 0, ρ3 · X = 0
c(ρ1) = x, c(ρ2) = y
2, c(ρ3) = yw
where c is the map induced by the map Z → F2 from the exact sequence (51).
Proof. The strategy of the proof is to consider four exact couples induced by the exact sequence (51):
H∗(D8,Z)
×2 ✲ H∗(D8,Z) H
∗(H1,Z)
×2 ✲ H∗(H1,Z)
H∗(D8,F2)
H∗(H1,F2)
H∗(H2,Z)
×2 ✲ H∗(H2,Z) H
∗(K4,Z)
×2 ✲ H∗(K4,Z)
H∗(H2,F2)
H∗(K4,F2)
and the corresponding morphisms induced by resD8H1 , res
and resD8K4 . Our notation is as in the restriction
diagram (27).
1. The module Z as a H1-module is a trivial module. Therefore in the H1 exact couple d1 is the usual
Bockstein homomorphism and so
d1(a) = Sq
1(a) = a2, d1(b) = Sq
1(b) = b2.
Thus from the restriction homomorphism resD8H1 we have:
resD8H1 (d1(1)) = d1
resD8H1 (1)
= d1(1) = 0 ⇒ d1(1) ∈ ker
resD8H1
resD8H1 (d1(x)) = d1
resD8H1 (x)
= d1(0) = 0 ⇒ d1(x) ∈ ker
resD8H1
resD8H1 (d1(y)) = d1
resD8H1 (y)
= d1(b) = b
2 ⇒ d1(y) ∈ y
2 + ker
resD8H1
resD8H1 (d1(w)) = d1
resD8H1 (w)
= ba(a+ b) ⇒ d1(w) ∈ yw + ker
resD8H1
2. The module Z as a H2-module is a non-trivial module. The H2 ∼= Z4 exact couple unrolls into a
long exact sequence [8, Proposition 6.1, page 71]
H0(Z4,Z)
×2✲ H
0(Z4,Z)
0(Z4,F2)
F2 , 1
1(Z4,Z)
F2 , λ
×2✲ H
1(Z4,Z)
F2 , λ
1(Z4,F2)
F2 , e
2(Z4,Z)
×2✲ H
2(Z4,Z)
2(Z4,F2)
F2 , u
3(Z4,Z)
F2 , λU
×2✲ H
3(Z4,Z)
F2 , λU
3(Z4,F2)
F2 , eu
4(Z4,Z)
. . .
Here we have used the facts that Hi(Z4,Z) =
F2 , i odd
0 , i even
and that multiplication by U ∈
H2(Z4,Z) in H
∗(Z4,Z) is an isomorphism [11, Section XII. 7. pages 250-253]. The long exact
sequence describes the boundary operator:
δ(1) = λ, δ(e) = 0, δ(u) = λU
and consequently the first differential:
d1(1) = e, d1(e) = 0, d1(u) = eu.
The restriction homomorphism resD8H2 implies that:
resD8H2 (d1(1)) = d1
resD8H1 (1)
= d1(1) = e ⇒ d1(1) ∈ x+ ker
resD8H2
resD8H1 (d1(x)) = d1
resD8H1 (x)
= d1(e) = 0 ⇒ d1(x) ∈ ker
resD8H2
resD8H1 (d1(y)) = d1
resD8H1 (y)
= d1(e) = 0 ⇒ d1(y) ∈ ker
resD8H2
resD8H1 (d1(w)) = d1
resD8H1 (w)
= d1(u) = eu ⇒ d1(w) ∈ yw + ker
resD8H2
3. The module Z as a K4-module is a non-trivial module. Then the K4 ∼= Z2 exact couple unrolls into
H0(Z2,Z)
×2✲ H
0(Z2,Z)
0(Z2,F2)
F2 , 1
1(Z2,Z)
F2 , ϕ
×2✲ H
1(Z2,Z)
F2 , ϕ
1(Z2,F2)
F2 , t4
2(Z2,Z)
×2✲ H
2(Z2,Z)
2(Z2,F2)
F2 , t
3(Z2,Z)
F2 , ϕT
×2✲ H
3(Z2,Z)
F2 , ϕT
3(Z2,F2)
F2 , t
4(Z2,Z)
. . .
Similarly, Hi(Z2,Z) =
F2 , i odd
0 , i even
and multiplication by T ∈ H2(Z2,Z) in H
∗(Z2,Z) is an
isomorphism [11, Section XII. 7. pages 250-253]. Then
d1(1) = t4, d1(t
4 ) = 0, d1(t
4 ) = t
for i ≥ 0. This implies that
resD8K4 (d1(w)) = d1
resD8H1 (w)
= d1(0) = 0 (55)
resD8K4 (d1(y)) = d1
resD8H1 (y)
= d1(0) = 0. (56)
From (52), (53) and the restriction diagram (27) follows:
d1(1) = x, d1(x) = 0,
d1(w) ∈ {yw, yw + x
3} and d1(y) ∈ {y
2, y2 + x2}. (57)
Since resD8K4 (yw) = 0, res
yw + x3
= t34 6= 0 and res
= 0, resD8K4
y2 + x2
= t24 6= 0, then the
equations (55) and (56) resolve the final dilemmas (57). Thus d1(w) = yw. According to [10, Remark
5.7.4, page 108] there are elements ρ1, ρ2, ρ3 of degree 1, 2, 3 and of exponent 2 in H
∗(D8,Z) satisfying
property (B) of this proposition.
The property (A) follows from the properties of Bockstein spectral sequence and the fact that the
derived couple of the D8 exact couple is:
0 ✲ 0
where F2 appears in dimension 0.
Remark 6.2. The proposition does not describes the completeH∗(D8,Z)-modulo structure onH
∗(D8,Z).
It gives only the necessary information for the computation of IndexD8,ZS(R
4 ). The complete result
can be found in [18, Theorem 5.11.(a)].
Thus, the index is given by
IndexD8,ZS(R
4 ) = 〈∂
1,3j−1
3j,Z (ρ1), ∂
2,3j−1
3j,Z (ρ2), ∂
3,3j−1
3j,Z (ρ3)〉.
The morphism C from spectral sequence (50) to spectral sequence (44) induced by the reduction homo-
morphism Z→ F2 implies that:
1,3j−1
3j,Z (ρ1)) = ∂
1,3j−1
3j,F2
(c∗(ρ1)) = ∂
1,3j−1
3j,F2
(x) = 0
2,3j−1
3j,Z (ρ2)) = ∂
2,3j−1
3j,F2
(c∗(ρ2)) = ∂
2,3j−1
3j,F2
(y2) = yj+2wj = yj+1wj−1(y + x)w
3,3j−1
3j,Z (ρ3)) = ∂
3,3j−1
3j,F2
(c∗(ρ3)) = ∂
3,3j−1
3j,F2
(yw) = yj+1wj+1
The sequence of D8 inclusion maps
⊕(j−1)
4 ) ⊂ S(R
4 ) ⊂ S(R
⊕(j+1)
provides (Proposition 3.2) a sequence of inclusions:
2 〉 = IndexD8,ZS(R
⊕(j−1)
4 ) ⊇ IndexD8,ZS(R
4 ) ⊇ IndexD8,ZS(R
⊕(j+1)
4 ) = 〈Y
2 〉. (59)
The relations (58), (59) and (40), along with Proposition 6.1 imply that for j odd:
IndexD8,ZS(R
4 ) = 〈Y
2 M,Y
Remark 6.3. The index IndexD8,ZS(Uk × R
4 ) appearing in the join test map scheme can now be
computed. From Example 3.4 and the restriction diagram (43) it follows that
IndexD8,ZS(Uk) = IndexD8,ZD8/H1 = ker
resD8H1 : H
∗(D8,Z)→ H
∗(H1,Z)
= 〈X 〉.
The inclusions
IndexD8,ZS(Uk ×R
4 ) ⊆ IndexD8,ZS(R
4 ) and IndexD8,ZS(Uk ×R
4 ) ⊆ IndexD8,ZS(Uk)
imply that
IndexD8,ZS(Uk ×R
4 ) ⊆ IndexD8,ZS(R
4 ) ∩ IndexD8,ZS(Uk) = {0}.
Thus, as in the case of F2 coefficients, the Fadell–Husseini index theory with Z coefficients on the join
CS/TM scheme does not lead to any obstruction to the existence of the equivariant map in question.
7 IndexD8,F2S
This section is devoted to the proof of the equality
IndexD8,F2S
d × Sd = 〈πd+1, πd+2, w
d+1〉. (60)
The index will be determined by the explicit computation of the Serre spectral sequence associated with
the Borel construction
Sd × Sd → ED8 ×D8
Sd × Sd
→ BD8.
The group D8 acts nontrivially on the cohomology of the fibre, and therefore the spectral sequence has
nontrivial local coefficients. The E2-term is given by
2 = H
p(BD8,H
q(Sd × Sd,F2)) = H
p(D8, H
q(Sd × Sd,F2))
Hp(D8,F2) , q = 0, 2d
Hp(D8,F2[D8/H1]) , q = d
0 , q 6= 0, d, 2d.
The nontriviality of local coefficients appears in the d-th row of the spectral sequence.
In Section 7.4 there is a sketch of an alternative proof of the fact (60) suggested by a referee for an
earlier, F2-coefficient, version of the paper.
7.1 The d-th row as an H∗(D8,F2)-module
Since the spectral sequence is an H∗(D8,F2)-module and the differentials are module maps we need to
understand the H∗(D8,F2)-module structure of the E2-term.
Proposition 7.1. H∗(D8,F2[D8/H1]) ∼=ring H
∗(H1,F2).
Proof. Here H1 = 〈ε1, ε2〉 ∼= Z2 × Z2 is a maximal (normal) subgroup of index 2 in D8.
Method 1: The statement follows from Shapiro’s lemma [8, Proposition 6.2, page 73] and the fact that
when [G : H ] <∞, then there is an isomorphism of G-modules CoindGHM
∼= IndGHM .
Method 2: There is an exact sequence of groups
1→ H1 → D8 → D8/H1 → 1.
The associated LHS spectral sequence [1, Corollary 1.2, page 116] has the E2-term:
2 = H
p(D8/H1, H
q(H1,F2[D8/H1]))
p(Z2, H
q((Z2)
2,F2 ⊕ F2))
∼= Hp(Z2;H
q((Z2)
2,F2)⊕H
q((Z2)
2,F2)).
The action of the group D8/H1 ∼= Z2 on the sum is given by the conjugation action of G on the pair
(H1, H
q(H1,F2[D8/H1])) [8, Corollary 8.4, page 80]. Since F2[Z2] is a free Z2-module
H0(Z2;F2[Z2]) = (F2[Z2])
Z2 = F2
and Hp(Z2;F2[Z2]) = 0 for p > 0. Thus
∼= Hp(D8/H1;H
q((Z2)
2,F2)⊕H
q((Z2)
2,F2))
p(D8/H1;F2[Z2]
∼= Hp(D8/H1;F2[Z2])
q+1 ∼=
Hp(Z2;F2[Z2])
)Z2 ∼= F
2 , p = 0
0 , p > 0.
Thus the E2-term has the shape as in Figure 5 (concentrated in the 0-column) and collapses.
The first information about the H∗(D8,F2)-module structure on H
∗(D8,F2[D8/H1]), as well as the
method for revealing the complete structure, comes from the following proposition.
ÝHDÝH1,F2Þ ã H
DÝH1,F2ÞÞ
D8/H1
0                                    1
Figure 5: The A2-term of the LHS spectral sequence
Proposition 7.2. We have x ·H∗(D8,F2[D8/H1]) = 0 for the nonzero element x ∈ H
1(D8,F2) that is
characterized by resD8H1(x) = 0.
Proof.
Method 1: The isomorphismH∗(D8,F2[D8/H1]) ∼=ring H
∗(H1,F2) induced by Shapiro’s lemma [8, Propo-
sition 6.2, page 73] carries the H∗(D8,F2)-module structure to H
∗(H1,F2) via the restriction homomor-
phism resD8H1 : H
∗(D8,F2)→ H
∗(H1,F2). In this way the complete H
∗(D8,F2)-module structure is given
on H∗(D8,F2[D8/H1]). In particular, since res
(x) = 0, the proposition is proved.
Method 2: The exact sequence of groups
1→ H1 → D8 → D8/H1 → 1
induces two LHS spectral sequences
2 = H
p (D8/H1, H
q (H1,F2[D8/H1])) =⇒ H
p+q(D8,F2[D8/H1]), (62)
2 = H
p (D8/H1, H
q (H1,F2)) =⇒ H
p+q(D8,F2). (63)
The spectral sequence (63) acts on the spectral sequence (62)
t → A
u+r,v+s
In the E∞-term this action becomes an action of H
∗(D8,F2) on H
∗(D8,F2[D8/H1]). Since we already
discussed both spectral sequences we know that
2 = A
∞ and B
2 = B
From Figures 2 and 5 it is apparent that x ∈ B
2 = B
∞ acts by x · A
2 = 0 for every p and q.
Corollary 7.3. Indexd+2D8,F2S
d × Sd = im
∂d+1 : E
d+1 → E
∗+d+1,0
⊆ y ·H∗(D8,F2).
Proof. Let α ∈ E
d+1 and ∂d+1(α) /∈ y ·H
∗(D8,F2). Then x · ∂d+1(α) 6= 0. Since ∂d+1 is a H
∗(D8,F2)-
module map and x acts trivially on H∗(D8,F2[D8/H1]), as indicated by Proposition 7.2, there is a
contradiction
0 = ∂d+1(x · α) = x · ∂d+1(α) 6= 0.
Proposition 7.4. H∗(D8,F2[D8/H1]) is generated as an H
∗(D8,F2)-module by
H0(D8,F2[D8/H1]) and H
1(D8,F2[D8/H1]).
Proof.
Method 1: We already observed that Shapiro’s lemma H∗(D8,F2[D8/H1]) ∼=ring H
∗(H1,F2) carries the
H∗(D8,F2)-module structure to H
∗(H1,F2) via the restriction homomorphism res
: H∗(D8,F2) →
H∗(H1,F2). Thus H
∗(H1,F2) as an H
∗(D8,F2)-module is generated by 1 ∈ H
0(H1,F2) together with
a ∈ H1(H1,F2).
Method 2: There is the exact sequence of D8-modules
0→ F2 → F2[D8/H1]→ F2 → 0, (64)
where the left and right modules F2 are trivial D8-modules. The first map is a diagonal inclusion while
the second one is a quotient map. The sequence (64) induces a long exact sequence on group cohomology
[8, Proposition 6.1, page 71],
0→ H0 (D8,F2)
→ H0 (D8,F2[D8/H1])
→ H0 (D8,F2)
H1 (D8,F2)
→ H1 (D8,F2[D8/H1])
→ H1 (D8,F2)
→ . . .
From the exact sequence (65), compatibility of the cup product [8, page 110, (3.3)] and Proposition 7.2 one
can deduce that δ0(1) = x. Then by chasing along sequence (65) with compatibility of the cup product [8,
page 110,(3.3)] as a tool it can be proved that H∗(D8,F2[D8/H1]) is generated as a H
∗(D8,F2)-module
by I = i0(1) and A ∈ q
1 ({y}).
7.2 Indexd+2D8,F2S
d × Sd = 〈πd+1, πd+2〉
The index by definition is
Indexd+2D8,F2S
d × Sd = im
∂d+1 : E
d+1 → E
∗+d+1,0
∂d+1 : H
∗ (D8,F2[D8/H1])→ H
∗+d+1(D8,F2
From Proposition 7.4 this image is generated as a module by the ∂d+1-images of H
0 (D8,F2[D8/H1])
and of H1 (D8,F2[D8/H1]). The ∂d+1 image is computed by applying restriction properties given in
Proposition 3.5 to the subgroup H1. With the identification of H
∗ (D8,F2[D8/H1]) given by Shapiro’s
lemma the morphism of spectral sequences of Borel constructions induced by restriction is specified in
Figure 6. Also,
Indexd+2D8,F2S
d × Sd = 〈∂D8d+1(1), ∂
d+1(a), ∂
d+1(b), ∂
d+1(a+ b)〉.
a + b
0 ^d+1
y^d+1
0 1 d + 1 d + 2
2d 11612
d 11ã12
a ã Ýa +bÞ
Ýa +bÞã a
b ã b
0 ad+1 + Ýa +bÞd+1
ad+2 + Ýa +bÞd+2
aÝa +bÞÝad + Ýa +bÞd Þ
bÝad+1 + Ýa +bÞd+1 Þ
0 1 2 3 4 d + 1 d + 2
Ed+1 term of the Borel construction
Sd × Sd ¸ ED8 ×D8 ÝS
d × Sd Þ ¸ BD8
term of the Borel construction
Sd × Sd ¸ EH1 ×H1 ÝS
d × Sd Þ ¸ BH1
Figure 6: The morphism of spectral sequences
To simplify notation let ρd := a
d + (a+ b)d+1. Then from
7−→ 11 ⊕ 12
7−→ ρd+1
{a, a+ b, b}
a⊕ (a+ b)
(a+ b)⊕ a
7−→ {ρd+2, a(a+ b)ρd, bρd+1}
it follows that
resD8H1
∂D8d+1(1), ∂
d+1(a), ∂
d+1(b) , ∂
d+1(a+ b)
= {ρd+2, a(a+ b)ρd, bρd+1}.
The formula
ρd+2 = a
d+2 + (a+ b)d+2
= (a+ a+ b)
ρd+1 + a(a+ b)
ai(a+ b)d−1−i
= bρd+1 + a(a+ b)(a+ a+ b)
ai(a+ b)d−1−i
= bρd+1 + a(a+ b)ρd
together with Remark 1.3 and the knowledge of the restriction resD8H1 implies that
resD8H1(πd) = ρd.
Therefore, there exist xα, xβ, xγ, xδ ∈ ker(resD8H1) such that
∂D8d+1(1) = πd+1 + xα
∂D8d+1(a), ∂
d+1(b), ∂
d+1(a+ b)
= {πd+2 + xβ, yπd+1 + xγ, wπd + xδ} .
Since y divides πd, Proposition 7.2 implies that α = β = γ = δ = 0, and
Indexd+2D8,F2S
d × Sd = 〈∂D8d+1(1), ∂
d+1(a), ∂
d+1(b), ∂
d+1(a+ b)〉
= 〈πd+1, πd+2, yπd+1, wπd〉
= 〈πd+1, πd+2〉.
Remark 7.5. The property that the concretely described homomorphism
resD8H1 : H
∗(D8,F2[D8/H1])→ H
∗(H1,F2[D8/H1])
is injective holds more generally [13, Lemma on page 187].
7.3 IndexD8,F2S
d × Sd = 〈πd+1, πd+2, w
In the previous section we described the differential ∂D8d+1 of the Serre spectral sequence associated with
the Borel construction
Sd × Sd → ED8 ×D8
Sd × Sd
→ BD8.
The only remaining, possibly non-trivial, differential is ∂D82d+1.
The following proposition describing E
2d+1 can be obtained from Figure 6.
Proposition 7.6. E
2d+1 = ker
∂D8d+1 : E
d+1 → E
∗+d+1,d
= x ·H∗(D8,F2)
Proof. The restriction property from Proposition 3.5(D), applied to the element 1 ∈ E
d+1 = H
∗(D8,F2)
implies that ∂D8d+1(1) 6= 0. Proposition 7.2, together with the fact that multiplication by y and by w in
H∗(D8,F2[D8/H1]) is injective, implies that ker
∂D8d+1 : E
d+1 → E
∗+d+1,d
= xH∗(D8,F2).
The description of the differential ∂D82d+1 : E
2d+1 → E
∗+2d+1,0
2d+1 comes in an indirect way. There is a
D8-equivariant map
Sd × Sd → Sd ∗ Sd ≈ S((V+− ⊕ V−+)
⊕(d+1)
given by Sd × Sd ∋ (t1, t2) 7→
t2 ∈ S
d ∗ Sd. The result of Section 5.1 and the basic property of
the index (Proposition 3.2) imply that
IndexD8,F2S
d × Sd ⊇ IndexD8,F2S((V+− ⊕ V−+)
⊕(d+1)
) = 〈wd+1〉.
Thus wd+1 ∈ IndexD8,F2S
d × Sd. Since by Corollary 7.3 wd+1 /∈ Indexd+1D8,F2S
d × Sd it follows that
wd+1 ∈ im
∂D82d+1 : E
2d+1 → E
2d+2,0
But the only nonzero element in E
2d+1 is x, therefore
∂D82d+1 (x) = w
This concludes the proof of equation (60).
7.4 An alternative proof, sketch
The objective of our index calculation is to find the kernel of the map (cf. Section 3)
H∗(ED8 ×D8
Sd × Sd
,F2) = H
(Sd × Sd,F2)← H
(pt,F2) = H
∗(ED8 ×D8 pt,F2). (66)
This map is induced by the map of spaces
ED8 ×D8 (S
d × Sd)→ ED8 ×D8 pt. (67)
From the definition of the product ×D8 the map (67) is induced by ED8 × (S
d × Sd) → ED8 × pt, i.e.
by (Sd × Sd)→ pt. The map (67), again by definition of product ×D8 is
ED8 × (S
d × Sd)
/D8 → (ED8 × pt) /D8. (68)
Let S2 ∼= Z2 denotes the quotient group D8/H1. There is a natural homeomorphisms [23, Proposition
1.59, page 40]
ED8 × (S
d × Sd)
/S2 → ((ED8 × pt) /H1) /S2 (69)
which is induced by the map
ED8 × (S
d × Sd)
/H1 → (ED8 × pt) /H1 (70)
Since ED8 is also a model for EH1, the map (70) is a projection map in the Borel construction of S
with respect to the group H1:
Sd × Sd ✲
ED8 × (S
d × Sd)
The group D8 acts freely on ED8 × (S
d × Sd) and on ED8 × pt. Therefore the S2 actions on the spaces(
ED8 × (S
d × Sd)
/H1 and (ED8 × pt) /H1 are also free. There are natural homotopy equivalences
ED8 × (S
d × Sd)
/S2 ≃ ES2 ×S2
ED8 × (S
d × Sd)
((ED8 × pt) /H1) /S2 ≃ ES2 ×S2 ((ED8 × pt) /H1)
which transform the map (69) into a map of Borel constructions
ES2 ×S2
ED8 × (S
d × Sd)
→ ES2 ×S2 ((ED8 × pt) /H1) (72)
induced by the map (70) on the fibres.
The map between Borel constructions (72) induces a map of associated Serre spectral sequences which
on the E2-term looks like
2 = H
p(S2, H
q((ED8 ×
Sd × Sd
)/H1,F2))← H
p(S2, H
q((ED8 × pt) /H1,F2)) = H
2 . (73)
The spectral sequence H
2 is the one studied in section 4.2. It converges to H
∗(D8,F2) and H
2 = H
Lemma 7.7. E
2 = E
Proof. The action of H1 on S
d × Sd is free. Therefore
ED8 ×
Sd × Sd
/H1 ≃
Sd × Sd
/H1 = RP
d × RP d (74)
where the induced action of S2 from
ED8 ×
Sd × Sd
/H1 onto RP
d × RP d interchanges the copies
of RP d × RP d. The S2-homotopy equivalence (74) induces an isomorphism of induced Serre spectral
sequences of Borel constructions
2 = H
p(S2, H
ED8 ×
Sd × Sd
/H1,F2)) ∼= H
p(S2, H
q(RP d × RP d,F2)) = G
Since for the spectral sequence G
2 , by [1, Theorem 1.7, page 118], we know that G
2 = G
∞ , the same
must hold for the spectral sequence E
We have obtained the following presentation of the map (66) and the related map of the fibres (70).
Proposition 7.8.
(A) The map H∗D8(pt,F2) → H
(Sd × Sd,F2) gives rise to a map of spectral sequences of S2-Borel
constructions
2 = H
p(S2, H
q((ED8 × pt) /H1,F2))→ H
p(S2, H
q((ED8 ×
Sd × Sd
)/H1,F2)) = E
2 (75)
which is induced by the map on fibres
ED8 × (S
d × Sd)
/H1 → (ED8 × pt) /H1.
(B) The map on the fibres is the projection map in the H1-Borel construction
Sd × Sd →
ED8 × (S
d × Sd)
/H1 → BH1 .
It is completely determined in F2 cohomology by its kernel:
H∗(H1,F2)→ H
ED8 × (S
d × Sd)
/H1,F2)
= IndexH1,F2S
d × Sd = 〈ad+1, (a+ b)d+1〉.
The E
2 = E
∞ andH
2 = H
∞ are described by [1, Lemma 1.4, page 117]. Therefore, IndexD8,F2S
or the kernel of the map of spectral sequences (75) is completely determined by the kernel of the map of
S2-invariants
F2[a, a+ b]
F2[a, a+ b]/〈a
d+1, (a+ b)d+1〉
H∗(H1,F2)
S2 → H∗(
ED8 × (S
d × Sd)
/H1,F2)
where S2 action is given by a 7−→ a+ b. The equation (60)
IndexD8,F2S
d × Sd = 〈πd+1, πd+2, w
d+1〉.
is a consequence of the previous discussion, identification of elements (23) in the spectral sequence (22)
and the following proposition about symmetric polynomials.
Proposition 7.9.
(A) A symmetric polynomial
aik(a+ b)jk ∈ F2[a, a+ b]
S2 is in the kernel of the map (76) if and only
if for every monomial
ad+1 | aik(a+ b)jk or (a+ b)d+1 | aik(a+ b)jk .
(B) The kernel of the map (76), as an ideal in F2[a, a+ b]
S2 is generated by
ad+1 + (a+ b)d+1, ad+2 + (a+ b)d+2, ad+1(a+ b)d+1.
The approach presented here, with all its advantages, has two disadvantages:
(1) The carrier of the combinatorial lower bound for the mass partition problem, the partial index
Indexd+2D8,F2S
d × Sd, cannot be obtained without extra effort.
(2) It cannot be used for computation of the index Indexd+2D8,ZS
d × Sd; the spectral sequence H
2 , if
considered with Z coefficients, is the sequence (35) whose E∞-term has a ring structure that differs
from H∗(D8,Z).
These were our reasons for presenting this idea just as a sketch.
8 IndexD8,ZS
Let Π0 = 0, Π1 = Y and Πn+2 = YΠn+1 +WΠn, for n ≥ 0, be a sequence of polynomials in H
∗(D8,Z).
This section is devoted to the proof of the equality
Indexd+2D8,ZS
d × Sd =
〈Π d+2
,Π d+4
,MΠ d
〉 , for d even
〈Π d+1
,Π d+3
〉 , for d odd.
The index is determined by the explicit computation of the Ed+2-term of the Serre spectral sequence
associated with the Borel construction
Sd × Sd → ED8 ×D8
Sd × Sd
→ BD8.
As in the previous section, the group D8 acts nontrivially on the cohomology of the fibre and thus the
coefficients in the spectral sequence are local. The E2-term is given by
2 = H
p(BD8,H
q(Sd × Sd,Z)) = Hp(D8, H
q(Sd × Sd,Z))
Hp(D8,Z) , q = 0, 2d
Hp(D8, H
d(Sd × Sd,Z)) , q = d
0 , q 6= 0, d, 2d.
The local coefficients are nontrivial in the d-th row of the spectral sequence.
8.1 The d-th row as an H∗(D8,Z)-module
The D8-module M := H
d(Sd × Sd,Z), as an abelian group, is isomorphic to Z× Z. Since the action of
D8 on M depends on d we distinguish two cases.
8.1.1 The case when d is odd
The action on M is given by
ε1 · (x, y) = (x, y), ε2 · (x, y) = (x, y), σ · (x, y) = (y, x).
Thus, there is an isomorphism of D8-modules M ∼= Z[D8/H1]. The situation resembles the one in Section
7.1, and therefore the following propositions hold.
Proposition 8.1. H∗(D8,Z[D8/H1]) ∼=ring H
∗(H1,Z).
Proof. The claim follows from Shapiro’s lemma [8, Proposition 6.2, page 73] and the fact that when
[G : H ] <∞ there is an isomorphism of G-modules CoindGHM
∼= IndGHM .
Proposition 8.2. Let T ∈H∗(D8,Z) and P ∈ H
∗(H1,Z) ∼= H
∗(D8,Z[D8/H1]).
(A) The action of H∗(D8,Z) on H
∗(D8,Z[D8/H1]) is given by
T ·P := resD8H1 (T ) ·P .
Here P on the right hand side is an element of H∗(H1,Z) and on the left hand side is its isomorphic
image under the isomorphism from the previous proposition. In particular,
X·H∗(D8,Z[D8/H1]) = 0.
(B) H∗(D8,Z)-module H
∗(D8,Z[D8/H1]) is generated by the two elements
1, α ∈ H∗(H1,Z) ∼= H
∗(D8,Z[D8/H1])
of degree 0 and 2.
(C) The map H∗(D8,Z[D8/H1])→ H
∗(D8,F2[D8/H1]), induced by the coefficient map Z→ F2, is given
by 1, α 7−→ 1, a2.
Proof. The isomorphism H∗(D8,Z[D8/H1]) ∼=ring H
∗(H1,Z) induced by Shapiro’s lemma [8, Propo-
sition 6.2, page 73] carries the H∗(D8,Z)-module structure to H
∗(H1,Z) via res
: H∗(D8,Z) →
H∗(H1,Z). In this way the complete H
∗(D8,Z)-module structure is given on H
∗(D8,Z[D8/H1]). The
claim (B) follows from the restriction diagram (43). The morphism of restriction diagrams induced by
the coefficient reduction homomorphism c : Z→ F2 implies the last statement.
8.1.2 The case when d is even
The action on M is given by
ε1 · (x, y) = (−x, y), ε2 · (x, y) = (x,−y), σ · (x, y) = (y, x).
In this case we are forced to analyze the Bockstein spectral sequence associated with the exact sequence
of D8-modules
→M → F2[D8/H1]→ 0, (79)
i.e. with the exact couple
H∗(D8,M)
×2 ✲ H∗(D8,M)
H∗(D8,F2[D8/H1]).
First we study the Bockstein spectral sequence
H∗(H1,M)
×2 ✲ H∗(H1,M)
H∗(H1,F2[D8/H1]).
As in Section 7.2, we have that H∗(H1,F2[D8/H1]) = F2[a, a + b] ⊕ F2[a, a + b]. The module M as an
H1-module can be decomposed into the sum of two H1-modules Z1 and Z2. The modules Z1 ∼=Ab Z and
Z2 ∼=Ab Z are given by
ε1 · x = −x, ε2 · x = x and ε1 · y = y, ε2 · y = −y
for x ∈ Z1 and y ∈ Z2. This decomposition also induces a decomposition of H1-modules F2[D8/H1] ∼=
F2 ⊕ F2. Thus, the exact couple (81) decomposes into the direct sum of two exact couples
H∗(H1, Z1)
×2 ✲ H∗(H1, Z1) H
∗(H1, Z2)
×2 ✲ H∗(H1, Z2)
H∗(H1,F2)
H∗(H1,F2)
Since all the maps in these exact couples are H∗(H1,Z)-module maps, the following proposition com-
pletely determines both exact couples.
Proposition 8.3. In the exact couples (82) differentials d1 = c ◦ δ are determined, respectively, by
d1(1) = a, d1(b) = b(b+ a) and d1(1) = a+ b, d1(a) = d1(b) = ab. (83)
Proof. In both claims we use the following diagram of exact couples induced by restrictions, where
i ∈ {1, 2}:
(H1, Zi)
×2 ✲ H∗(H1, Zi)
(H1,F2)
(K1, Zi)
×2 ✲ H∗(K1, Zi)
(K1,F2)
(K2, Zi)
×2 ✲ H∗(K2, Zi)
(K2,F2)
(K3, Zi)
×2 ✲ H∗(K3, Zi)
(K3,F2)
The first exact couple. The module Z1 is a non-trivial K1 and K3-module, but a trivial K2-module.
Therefore by the long exact sequences (54), properties of Steenrod squares and the assumption at the
end of the Section 4.3.2:
(A) K1-exact couple: d1(1) = t1 and d1(t1) = 0;
(B) K2-exact couple: d1(1) = 0 and d1(t2) = t
(C) K3-exact couple: d1(1) = t3 and d1(t3) = 0.
resH1K1(d1(1)) = t1
resH1K2(d1(1)) = 0
resH1K3(d1(1)) = t3
⇒ d1(1) = a
resH1K1(d1(b)) = 0
resH1K2(d1(b)) = t
resH1K3(d1(b)) = 0
⇒ d1(b) = b(b+ a).
The second exact couple. The module Z2 is a non-trivial K2 and K3-module, while it is a trivial K1-
module. Therefore by the long exact (54), properties of Steenrod squares and the assumption at the end
of the Section 4.3.2:
(A) K1-exact couple: d1(1) = 0 and d1(t1) = t
(B) K2-exact couple: d1(1) = t2 and d1(t2) = 0;
(C) K3-exact couple: d1(1) = t3 and d1(t3) = 0.
resH1K1(d1(1)) = 0
resH1K2(d1(1)) = t2
resH1K3(d1(1)) = t3
⇒ d1(1) = a+ b
resH1K1(d1(b)) = t
resH1K2(d1(b)) = 0
resH1K3(d1(b)) = 0
⇒ d1(b) = ab.
Remark 8.4. The result of the previous proposition can be seen as a key step in an alternative proof of
the equation (20).
Proposition 8.5. In the exact couple (80), with identification H∗(D8,F2[D8/H1]) = F2[a, a + b], the
differential d1 = s ◦ δ satisfies
d1(1) = a, d1(a+ b) = d1(b) = b(b+ a), d1(a
2) = a3. (84)
(This determines d1 completely since c and δ are H
∗(D8,Z)-module maps.)
Proof. Recall from the Remark 7.5 that the restriction map
resD8H1 : H
∗(D8,F2[D8/H1])→ H
∗(H1,F2[D8/H1])
is injective. Then the equations (84) are obtained by filling the empty places in the following diagrams
d1 ✲ a+ b
d1 ✲ a2
❄ d1✲ a⊕ (a+ b)
(a+ b)⊕ a
❄ d1✲ b(b+ a)⊕ ab
a2 ⊕ (a+ b)2
d1✲ a3 ⊕ (a+ b)3
where all vertical maps are resD8H1 .
Corollary 8.6. H∗(D8,M) is generated as a H
∗(D8,Z)-module by three elements ζ1, ζ2, ζ3 of degree 1,
2, 3 such that
c(ζ1) = a, c(ζ2) = b(a+ b), c(ζ3) = a
where c is the map H∗(D8,M)→ H
∗(D8,F2[D8/H1]) from the exact couple (80).
8.2 Indexd+2D8,ZS
d × Sd
The relation between the sequences of polynomials πd ∈ H
∗(D8,F2) and Πd ∈ H
∗(D8,Z) is described by
the following lemma.
Lemma 8.7. Let c∗ : H
∗(D8,Z)→ H
∗(D8,F2) be the map induced by the coefficient morphism Z→ F2
(explicitly given by (40)). Then for every d ≥ 0,
c∗(Πd) = π2d.
Proof. Induction on d ≥ 0. For d = 0 and d = 1 the claim is obvious. Let d ≥ 2 and let us assume that
claim holds for every d ≤ k + 1. Then
c∗(Πk+2) = c∗(YΠk+1 +WΠk)
hypo.
= y2π2k+2 + w
2π2k = y
2π2k+2 + ywπ2d+1 + ywπ2d+1 + w
= y(yπ2k+2 + wπ2d+1) + w(yπ2d+1 + wπ2k) = yπ2k+3 + wπ2k+2
= π2k+4.
There is a sequence of D8-inclusions
S1 × S1 ⊂ S2 × S2 ⊂ · · · ⊂ Sd−1 × Sd−1 ⊂ Sd × Sd ⊂ Sd+1 × Sd+1 ⊂ · · ·
implying a sequence of ideal inclusions
Index
S1 × S1 ⊇ Index4D8,ZS
2 × S2 ⊇ · · · ⊇ Indexd+1D8,ZS
d−1 × Sd−1 ⊇ Indexd+2D8,ZS
d × Sd ⊇ · · · (85)
8.2.1 The case when d is odd
In this section we prove that
Index
Sd × Sd = 〈Π d+1
,Π d+3
〉. (86)
The proof can be conducted as in the case of F2 coefficients (Section 7.2). The results of Section 7.2
can also be used to simplify the proof of equation (86). The morphism c∗ : H
∗(D8,Z) → H
∗(D8,F2)
induced by the coefficient morphism Z → F2 is a part of the morphism C of Serre spectral sequences
(78) and (61). Thus, for 1 ∈ E
d+1 = H
0(D8, H
d(Sd × Sd,Z)), 1̂ ∈ E0,dd+1 = H
0(D8, H
d(Sd × Sd,F2)),
α ∈ E
d+1 = H
2(D8, H
d(Sd × Sd,Z)) and a ∈ E1,dd+1 = H
1(D8, H
d(Sd × Sd,Z)),
C(∂d+1(1)) = ∂d+1(C(1)) = ∂d+1(1̂) = πd+1 = C
Π d+1
C(∂d+1(α)) = ∂d+1(C(α)) = ∂d+1(a
2) = ∂d+1(w · 1̂ + y · a) = wπd+1 + yπd+2 = πd+3 = C
Π d+3
From Proposition 8.2 and the sequence of inclusions (85) it follows that
∂d+1(1) = Π d+1
and ∂d+1(α) = Π d+3
Finally, the statement (B) of Proposition 8.2 implies equation (86).
8.2.2 The case when d is even
In this section we prove that
Indexd+2D8,ZS
d × Sd = 〈Π d+2
,Π d+4
,MΠ d
〉. (87)
The previous section implies that
,Π d+2
〉 ⊇ Indexd+2D8,ZS
d × Sd ⊇ 〈Π d+2
,Π d+4
〉. (88)
From Corollary 8.6 we know that Indexd+2D8,ZS
d × Sd is generated by three elements ∂d+1(ζ1), ∂d+1(ζ2),
∂d+1(ζ3) of degrees d + 2, d + 3, d + 4. Thus, ∂d+1(ζ1) = Π d+2
and ∂d+1(ζ2) = MΠ d
. Since Π d+4
〈Π d+2
,MΠ d
〉, then ∂d+1(ζ3) = Π d+4
. The proof of the equation (87) is concluded.
Alternatively, the proof can be obtained with the help of the morphism C of Serre spectral sequences
(78) and (61).
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http://arxiv.org/abs/1004.0746
	1 Introduction
	1.1 The hyperplane mass partition problem
	1.2 Statement of the main result (k=2)
	1.3 Proof overview
	1.4 Evaluation of the index bounds
	1.4.1 F2-evaluation
	1.4.2 Z-evaluation
	2 Configuration space/Test map scheme
	2.1 Configuration space
	2.2 Test map
	2.3 The test space
	3 The Fadell–Husseini index theory
	3.1 Equivariant cohomology
	3.2 IndexG,R and IndexG,Rk
	3.3 The restriction map and the index
	3.4 Basic calculations of the index
	3.4.1 The index of a product
	3.4.2 The index of a sphere
	4 The cohomology of D8 and the restriction diagram
	4.1 The poset of subgroups of D8
	4.2 The cohomology ring H(D8,F2)
	4.3 The cohomology diagram of subgroups with coefficients in F2
	4.3.1 The Z2Z2-diagram
	4.3.2 The D8-diagram
	4.4 The cohomology ring H(D8,Z)
	4.4.1 Evens' view
	4.4.2 The Bockstein spectral sequence view
	4.5 The D8-diagram with coefficients in Z
	4.5.1 The Z2Z2-diagram
	4.5.2 The D8-diagram
	5 IndexD8,F2S(R4j)
	5.1 IndexD8,F2S(V-+V+-)="426830A w"526930B 
	5.2 IndexD8,F2S(V–)="426830A y"526930B 
	5.3 IndexD8,F2S(R4j)="426830A yjwj"526930B 
	6 IndexD8,ZS(R4j)
	6.1 The case when j is even
	6.2 The case when j is odd
	7 IndexD8,F2SdSd
	7.1 The d-th row as an H(D8,F2)-module
	7.2 IndexD8,F2d+2SdSd="426830A d+1,d+2"526930B 
	7.3 IndexD8,F2SdSd="426830A d+1,d+2,wd+1"526930B 
	7.4 An alternative proof, sketch
	8 IndexD8,Z SdSd
	8.1 The d-th row as an H(D8,Z )-module
	8.1.1 The case when d is odd
	8.1.2 The case when d is even
	8.2 IndexD8,Zd+2SdSd
	8.2.1 The case when d is odd
	8.2.2 The case when d is even
