Smooth maps with singularities of bounded K-codimensions
Smooth maps with singularities of bounded
K-codimensions ∗†
Yoshifumi ANDO
Abstract
Let N and P be smooth manifolds of dimensions n and p respectively
such that n ≧ p ≧ 2 or n < p. Let Oℓ(N, P ) denote a K-invarinat open
subspace of J∞(N, P ) which consists of all regular jets and singular jets
z with codimKz ≦ ℓ (including fold jets if n ≧ p). An Oℓ-regular map
f : N → P refers to a smooth map such that j∞f(N) ⊂ Oℓ(N, P ). We will
prove that a continuous section s of Oℓ(N,P ) over N has an Oℓ-regular
map f such that s and j∞f are homotopic as sections. We next study
the filtration of the group of homotopy self-equivalences of a manifold P
which is constructed by the sets of Oℓ-regular homotopy self-equivalences
for nonnegative integers ℓ.
1 Introduction
Let N and P be smooth (C∞) manifolds of dimensions n and p respectively. Let
Jk(N,P ) denote the k-jet space of the manifolds N and P with the projections
πkN and π
P onto N and P mapping a jet onto its source and target respectively.
The canonical fiber is the k-jet space Jk(n, p) of C∞-map germs (Rn, 0) →
(Rp, 0). Let K denote the contact group defined in [MaIII]. Let O(n, p) denote
a K-invariant nonempty open subset of Jk(n, p) and let O(N,P ) denote an
open subbundle of Jk(N,P ) associated to O(n, p). In this paper a smooth map
f : N → P is called an O-regular map if jkf(N) ⊂ O(N,P ).
We will study what is called the homotopy principle for O-regular maps.
As for the long history of the several types of homotopy principles and their
applications we refer to the Smale-Hirsch Immersion Theorem ([Sm] and [H]),
the Feit k-mersion Theorem ([F]), the Phillips Submersion Theorem ([P]) and
the general theorems due to Gromov ([G1]) and du Plessis ([duP1], [duP2] and
[duP3]). Furthermore, we should refer to the homotopy principle on the 1-jet
level for fold-maps due to Èliašberg ([E1] and [E2]) (see further references in
[G2]).
∗2000 Mathematics Subject Classification. Primary 58K30; Secondary 57R45, 58A20
†Key Words and Phrases: smooth map, singularity, homotopy principle
‡This research was partially supported by Grand-in-Aid for Scientific Research (No.
16540072).
http://arxiv.org/abs/0704.0115v2
Let C∞
(N,P ) denote the space consisting of all O-regular maps, N → P
equipped with the C∞-topology. Let ΓO(N,P ) denote the space consisting of all
continuous sections of the fiber bundle πkN |O(N,P ) : O(N,P ) → N equipped
with the compact-open topology. Then there exists a continuous map jO :
(N,P ) → ΓO(N,P ) defined by jO(f) = j
kf . If the following property (h-P)
holds, then we say in this paper that the relative homotopy principle on the
existence level holds for O-regular maps.
(h-P) Let C be a closed subset of N with ∂N = ∅. Let s be a section in
ΓO(N,P ) which has an O-regular map g defined on a neighborhood of C to P ,
where jkg = s. Then there exists an O-regular map f : N → P such that s
and jkf are homotopic relative to a neighborhood of C by a homotopy sλ in
ΓO(N,P ) with s0 = s and s1 = j
As important applications of [An7, Theorem 0.1] we will prove the following
relative homotopy principles in (h-P). Here, Σn−p+1,0(n, p) refers to the space
consisting of all fold jets in Jk(n, p).
Theorem 1.1 Let n and p be positive integers with n ≧ p ≧ 2 or n < p. Let k
be a positive integer with k ≧ n− |n− p|+ 2. Let O(n, p) denote a K-invariant
open subspace of Jk(n, p) containing all regular jets such that if n ≧ p ≧ 2,
then O(n, p) contains Σn−p+1,0(n, p) at least. Let N and P be connected smooth
manifolds of dimensions n and p respectively with ∂N = ∅. Let C be a closed
subset of N . Let s be a section in ΓO(N,P ) which has an O-regular map g
defined on a neighborhood of C to P , where jkg = s.
Then there exists an O-regular map f : N → P such that jkf is homotopic
to s relative to a neighborhood of C as sections in ΓO(N,P ).
Let ρ be an integer with ρ ≧ 1. Let W kρ denote the subset consisting of
all z ∈ Jk(n, p) such that the codimension of Kz in Jk(n, p) is not less than
ρ (k may be ∞). Let Okℓ (n, p) denote a K-invariant nonempty open subset of
Jk(n, p)\W kℓ+1. By applying Theorem 1.1 we will prove the following theorem.
Theorem 1.2 Let ℓ be a positive integer. Let k ≧ max{ℓ+1, n−|n−p|+2} or
k = ∞. Let Okℓ (n, p) denote a K-invariant open subspace of J
k(n, p) containing
all regular jets such that if n ≧ p ≧ 2, then Okℓ (n, p) contains Σ
n−p+1,0(n, p) at
least. Then the relative homotopy principle in (h-P) holds for Okℓ -regular maps.
It is well known that any smooth map f : N → P is homotopic to a smooth
map g : N → P such that j∞x g is of finite K-codimension for any x ∈ N (see,
for example, [W, Theorem 5.1]).
There have been described many important applications of the homotopy
principles in [G2]. We only refer to the recent applications of the relative ho-
motopy principle on the existence level to the problems in topology such as the
elimination of singularities and the existence of Okl -regular maps in [An1-7] and
[Sa] and the relation between the stable homotopy groups of spheres and higher
singularities in [An4].
Let P be a closed manifold of dimension p. Let h(P ) denote the group of all
homotopy classes of homotopy equivalences of P . Let hℓ(P ) denote the subset
of h(P ) which consists of all homotopy classes of maps which are homotopic
to O∞l -regular homotopy equivalences. In particular, h0(P ) is the subset of all
homotopy classes of maps which are homotopic to diffeomorphisms of P . In this
paper we will prove that the following filtration
h0(P ) ⊂ h1(P ) ⊂ · · · ⊂ hℓ(P ) ⊂ · · · ⊂ h(P ). (1.1)
is never trivial in general.
Theorem 1.3 For a given positive integer d, there exists a closed oriented p-
manifold P and a sequence of positive integers ℓ1, ℓ2, · · · , ℓd with ℓj < ℓj+1 for
1 ≤ j < d such that
h0(P ) & hℓ1(P ) & hℓ2(P ) & · · · & hℓd(P ) & h(P ).
In Section 2 we will review the results on the Boardman manifolds and the
fundamental properties of K-equivalence and K-determinacy which are neces-
sary in this paper. In Section 3 we will recall [An7, Theorem 0.1] and apply it in
the proofs of Theorems 1.1 and 1.2. In Section 4 we will study the nonexistence
problem of Okl -regular maps. In Section 5 we will study the filtration in (1.1)
and prove Theorem 1.3.
2 Boardman manifolds and K-orbits
Throughout the paper all manifolds are Hausdorff, paracompact and smooth of
class C∞. Maps are basically smooth (of class C∞) unless otherwise stated.
For a Boardman symbol (simply symbol) I = (i1, · · · , ik) with n ≧ i1 ≧ · · · ≧
ik ≧ 0, let Σ
I(n, p) denote the Boardman manifold of symbol I in Jk(n, p) which
has been defined in [T], [L], [Bo] and [MaTB]. Let An = R[[x1, · · · , xn]] denote
the formal power series of algebra on variables x1, · · · , xn. Letmn be its maximal
ideal and An(k) = An/m
n . Let z = j
0 f ∈ J
k(n, p) where f = (f1, · · · , fp) :
(Rn, 0) → (Rp, 0). We define I(z) to be the ideal in An(k) generated by the
image in An(k) of the Taylor expansions of f
1, · · · , fp. It has been proved in
[Bo] and [MaTB] that the Boardman symbol I(z) of z depends only on the
ideal I(z) by the notion of the Jacobian extension. Let ΣI(N,P ) denote the
subbundle of Jk(N,P ) overN×P associated to ΣI(n, p). Let ΣIx,y(N,P ) denote
the fiber of ΣI(N,P ) over (x, y) ∈ N × P .
Since codimΣi1(n, p) = (p−n+ i1)i1, the following proposition follows from
[An6, Remark 2.1], which has been proved by using the results in [Bo, Section
Proposition 2.1 Let I = (i1, · · · , iℓ) be a symbol such that i1 ≧ max{n− p+
1, 1} and ΣI(n, p) is nonempty. Then we have
codimΣI(n, p) ≧ (p− n+ i1)i1 + (1/2)Σ
j=2 ij(ij + 1).
In particular, if iℓ > 0, then we have codimΣ
I(n, p) ≧ |n− p|+ ℓ.
Let ΩI(n, p) denote the union of all Boardman manifolds ΣJ (N,P ) with
J ≤ I in the lexicographic order. We have the following lemma (see [duP1]).
Lemma 2.2 The space ΩI(n, p) is open in Jk(n, p).
Let us review the K-equivalence of two smooth map germs f, g : (N, x) →
(P, y), which has been introduced in [MaIII, (2.6)], by following [Mart, II, 1].
We say that the above two map germs f and g are K-equivalent if there exists a
smooth map germ φ : (N, x) → GL(Rp) and a local diffeomorphism h : (N, x) →
(N, x) such that f(x) = φ(x)g(h(x)). It is known that this K-equivalence is
nothing but the contact equivalence introduced in [MaIII]. The contact group
K is defined as a certain subgroup of the group of germs of local diffeomorphisms
(N, x)× (P, y) and acts on Jkx,y(N,P ). For a k-jet z in J
x,y(N,P ) let Kz denote
the orbit of K through z. As is well known, Kz is an orbit of a Lie group. Hence,
Kz is a submanifold of Jkx,y(N,P ). This fact is also observed from the above
definition. The following lemma is important in this paper.
Lemma 2.3 The Boardman manifold ΣIx,y(N,P ) in J
x,y(N,P ) is invariant
with respect to the action of K.
Proof. Let z = jkxf and w = j
xg be k-jets in J
x,y(N,P ) such that two map
germs f and g are K-equivalent as above. Let h∗ : Cx → Cx be the isomorphism
defined by h∗(φ) = φ◦h. By the definition of K-equivalence we have h∗(I(g)) =
I(f). The Thom-Boardman symbols of jkxf and j
xg are determined by I(f)
and I(g), and are the same by [MaTB, 2, Corollary]. This proves the assertion.
Let us review the results in [MaIII], [MaIV] and [MaV] which are necessary
in this paper. Let C∞(N, x) and C∞(P, y) denote the rings of smooth function
germs on (N, x) and (P, y) respectively. Let mx and my denote their maximal
ideals respectively. Let f : (N, x) → (P, y) be a germ of a smooth map. Let
f∗ : C∞(P, y) → C∞(N, x) denote the homomorphism defined by f∗(a) = a◦f .
Let θ(N)x denote the C
∞(N, x)-module of all germs at x of smooth vector fields
on (N, x). We define θ(P )y similarly for y ∈ P . Let θ(f)x denote the C
∞(N, x)-
module of germs at x of smooth vector fields along f , namely which consists of
all smooth germs ς : (N, x) → TP such that pP ◦ ς = f . Here, pP : TP → P is
the canonical projection. Then we have the homomorphisms
tf : θ(N)x → θ(f)x (2.1)
defined by tf(uN) = df ◦ uN for uN ∈ θ(N)x. For a singular jet z = j
0 f ∈
Jk(N,P ) there has been defined the isomorphism
x,y(N,P )) −→ mxθ(f)x/m
x θ(f)x (2.2)
in [MaIII, (7.3)] such that Tz(Kz) corresponds to tf(mxθ(N)x)+f
∗(my)(θ(f)x)
modulo mk+1x θ(f)x. We do not here explain the definition. According to [MaIII]
we define d(f,K) to be
dimmxθ(f)x/(tf(mxθ(N)x) + f
∗(my)(θ(f)x)), (2.3)
which is equal to codimKz.
3 Proofs of Theorems 1.1 and 1.2.
In this section we prove Theorems 1.1 and 1.2.
Let k be a positive integer. Let W kρ = W
ρ (n, p) denote the subset consisting
of all z ∈ Jk(n, p) such that the codimension of Kz in Jk(n, p) is not less than
ρ. The following lemma has been observed in [MaV, Section 7 and Proof of
Theorem 8.1].
Lemma 3.1 Let ρ be an integer with ρ ≧ 1. Then W kρ is an algebraic subset of
Jk(n, p).
The order of K-determinacy is estimated by the codimension of a K-orbit as
follows.
Proposition 3.2 Let k be an integer with k > ρ. Let z = jkf be a singular jet
in Jk(n, p)\W kρ+1. Then z is K-k-determined.
Proof. It follows from [W, Theorem 1.2 (iii)] that if d =codimKz, then z
is K-(d + 1)-determined. Hence, if z ∈ Jk(n, p)\W kρ+1, then d ≤ ρ and z is
K-k-determined.
We define the bundle homomorphism
d : (πkN )
∗(TN) −→ (πkk−1)
∗(TJk−1(N,P )), (3.1)
d1 : (π
∗(TN) −→ (πkP )
∗(TP ).
Let w = jkxf ∈ J
x,y(N,P ) and z = π
k(w). Then we have j
k−1f : (N, x) →
(Jk−1(N,P ), z) and d(jk−1f) : TxN → Tz(J
k−1(N,P )). We set
dz(w,v) = (w, d(j
k−1f)(v)) and (d1)z(w,v) = (w, df(v)).
Let I ′ be a symbol of length k. Let K(ΣI
) denote the kernel subbundle of
(πkN |Σ
I′(N,P ))∗(TN) defined by
)w = (w,Ker(dxf)).
The following theorem follows from the corresponding assertion for the case
k = ∞ in [B, (7.7)]. This is very important in the proof of Theorem 1.1.
Theorem 3.3 If I ′ = (i1, · · · , ik−2, 0, 0) and I = (i1, · · · , ik−2,0), then we have
d(K(ΣI
)w) ∩ (π
k−1|Σ
I′(N,P ))∗(T (ΣI(N,P ))w = {0}
for any w ∈ ΣI
(N,P ).
Let us review a general condition on O(n, p) for the relative homotopy prin-
ciple on the existence level in [An7]. We say that a nonempty K-invariant open
subset O(n, p) is admissible if O(n, p) consists of all regular jets and a finite
number of disjoint K-invariant nonempty submanifolds V i(n, p) of codimension
ρi (1 ≤ i ≤ ι) such that the following properties (H-i) to (H-v) are satisfied.
(H-i) V i(n, p) consists of singular k-jets of rank ri, namely, V
i(n, p) ⊂
Σn−ri(n, p).
(H-ii) For each i, the set O(n, p)\{∪ιj=iV
j(n, p)} is an open subset.
(H-iii) For each i with ρi ≤ n, there exists a K-invariant submanifold
V i(n, p)(k−1) of Jk−1(n, p) such that V i(n, p) is open in (πkk−1)
−1(V i(n, p)(k−1)).
(H-iv) If n ≧ p ≧ 2, then V 1(n, p) = Σn−p+1,0(n, p).
Here, Σn−p+1,0(n, p) denotes the Thom-Boardman manifold in Jk(n, p), which
consists of K-orbits of fold jets. Let V i(N,P ) denote the subbundle of Jk(N,P )
associated to V i(n, p). Let K(V i) be the kernel bundle in (πkN )
∗(TN)|V i(N,P )
defined by K(V i)z = (z,Ker(dxf)).
(H-v) For each i with ρi ≤ n and any z ∈ V
i(N,P ), we have
d(K(V i)z) ∩ (π
k−1|V
i(N,P ))∗(T (V i(N,P )(k−1))z = {0}. (3.2)
Then we have proved the following theorem in [An7, Theorem 0.1].
Theorem 3.4 Let k ≧ n − |n − p| + 2. Let n ≧ p ≧ 2 or n < p. Let O(n, p)
denote an admissible open subspace of Jk(n, p). Then the relative homotopy
principle in (h-P) holds for O-regular maps.
We set
VI(n, p) = O(n, p) ∩ Σ
I(n, p).
Let J = (j1, · · · , jk) be a symbol of a singular jet with codimΣ
J (n, p) ≤ n. If
k ≧ n − |n − p|+ 2, we have by Proposition 2.1 that ik−1 = ik = 0. Indeed, if
ik−1 > 0, then
codimΣJ (n, p) ≧ |n− p|+ k − 1 ≧ n+ 1.
So we set J = (j1, · · · , jk−2, 0, 0), J
∗ = (i1, · · · , ik−2,0) and
VJ∗(n, p)
(k−1) = πkk−1(O(n, p)) ∩ Σ
J∗(n, p).
Lemma 3.5 Let J = (j1, · · · , jk−2, 0, 0) and J
∗ = (j1, · · · , jk−2,0) be as above.
Then VJ (n, p) is open in (π
−1(VJ∗(n, p)
(k−1)).
Proof. It is evident that
ΣJ(n, p) = (πkk−1)
−1(ΣJ
(n, p)) and O(n, p) ⊂ (πkk−1)
−1(πkk−1(O(n, p))).
So we have VJ (n, p) ⊂ (π
−1(VJ∗(n, p)
(k−1)). Since πkk−1 is an open map, we
have that VJ (n, p) is an open subset of (π
−1(VJ∗(n, p)
(k−1)).
Let us prove Theorem 1.1.
Proof of Theorem 1.1. By Theorem 3.4 it is enough to prove that O(n, p)
is admissible. Let J be a symbol of length k. By Lemma 2.3, VJ (n, p) is K-
invariant. We have that
(H1) O(n, p) is decomposed into a finite union of all VJ (n, p),
(H2) For each symbol J , the set O(n, p) ∩ ΩJ(n, p) is an open subset of
O(n, p),
(H3) VJ (n, p) is open in (π
−1(VJ∗(n, p)
(k−1)) by lemma 3.5,
(H4) If n ≧ p ≧ 2, then O(n, p) ⊃ Σn−p+1,0(n, p) by the assumption,
(H5) Property (3.2) holds for VJ (n, p) by Theorem 3.3 and Lemma 3.5.
Since O(n, p) satisfies the properties (H1) to (H5), we have proved Theorem
We next prove Theorem 1.2.
Proof of Theorem 1.2. If ℓ is finite, then it follows from Lemma 3.2 that if
k > ℓ, then any k-jet z of Jk(n, p)\W kℓ+1 is K-k-determined and we have
(π∞k )
−1(Okℓ (n, p)) = O
ℓ (n, p).
Therefore, if k ≧ max{ℓ+1, n−|n−p|+2}, then the relative homotopy principle
in (h-P) holds for Okℓ -regular maps by Theorem 1.1 and also for O
ℓ -regular
maps.
Corollary 3.6 Under the same assumption of Theorem 1.2, given a map f :
N → P is homotopic to an Okℓ -regular map if and only if there exists a section
s ∈ Γ
(N,P ) such that πkP ◦ s is homotopic to f .
Corollary 3.7 Let hℓ(P ) be as in Introduction. Then the homotopy class of a
homotopy equivalence f : P → P lies in hℓ(P ) if and only if j
∞f is homotopic
to a section in ΓO∞
(N,P ).
Here we give two remarks.
Remark 3.8 Let W∞
denote the subspace of J∞(n, p) which consists of all jets
z such that any smooth map germ f with z = j∞f is not finitely determined. Let
(N,P ) is the subbundle of J∞(N,P ) associated to W∞
. It has been proved
(see, for example, [W, Theorem 5.1]) that W∞
is not of finite codimension
in J∞(n, p). Consequently, the space of all smooth maps f : N → P with
j∞f(N) ⊂ J∞(N,P )\W∞
(N,P ) is dense in C∞(N,P ). In other words if N
is compact, then a smooth map f : N → P has an integer ℓ such that f is
homotopic to an O∞ℓ -regular map.
Remark 3.9 It is very important to study the topology of the space W kℓ+1(n, p)
and obstructions for finding an Okℓ -regular map. The Thom polynomials related
to W kℓ+1(n, p) have been studied in the dimensions n = p ≦ 8 in [O] and [F-R].
4 Nonexistence theorems
In this section we will discuss the nonexistence of Okℓ -regular maps f : N → P .
Let W kℓ+1(N,P ) denote the subbundle of J
k(N,P ) associated to W kℓ+1(n, p). By
the homotopy principle for Okℓ -regular maps in Theorem 1.2, the existence of
a section of Jk(N,P )\W kℓ+1(N,P ) over N is equivalent to the existence of an
Okℓ -regular map. However, it is not so easy to find obstructions associated to
W kℓ+1(N,P ) such as Thom polynomials of W
ℓ+1(N,P ), and so we will adopt a
method applied in [An1], [I-K] and [duP4] in this section.
For k ≧ p+1, let Σ(n, p; k) denote the algebraic subset of all C∞-nonstable
k-jets of Jk(n, p) defined in [MaV]. Note that for k′ > k, (πk
−1(Σ(n, p; k)) =
Σ(n, p; k′). We have proved the following proposition in [An1, Corollary 5.6].
Proposition 4.1 Let k ≧ p+ 1. If
(p− n+ i)(
i(i+ 1)− p+ n)− i2 ≧ n,
then we have that Σi(n, p) ⊂ Σ(n, p; k).
In [I-K] the following proposition has been proved, while it has not been
stated explicitly and the proof has been given in the context without the details.
So we give a sketchy proof.
Proposition 4.2 ([I-K]) Let ℓ be a nonnegative integer and k ≧ p+ ℓ+ 1. If
(p− n+ i)(
i(i+ 1)− p+ n)− i2 ≧ n+ ℓ,
then we have that Σi(n, p) ⊂ W kℓ+1(n, p). In particular, if n = p and
i2(i−1) ≧
n+ ℓ, then we have that Σi(n, n) ⊂ W kℓ+1(n, n).
Proof. Take a jet z in Σi(n, p) such that z = jk0 f . Suppose that z /∈ W
ℓ+1, and
hence codimKz ≦ ℓ. By [MaIV] there exists a versal unfolding F : (Rn×Rℓ, 0) →
(Rp × Rℓ, 0) of f and jk
(0,0)
F /∈ Σ(n + ℓ, p+ ℓ; k). Here, we note that jk
(0,0)
of kernel rank i. By the assumption and Proposition 4.1 we have
Σi(n+ ℓ, p+ ℓ) ⊂ Σ(n+ ℓ, p+ ℓ; k).
This implies jk
(0,0)
F ∈ Σ(n + ℓ, p+ ℓ; k). This is a contradiction. Hence, z lies
in W kℓ+1.
We show the following proposition by applying Proposition 4.2.
Proposition 4.3 Let ℓ be a nonnegative integer and k ≧ p+ℓ+1. If Σi(n, p) ⊂
W kℓ+1(n, p), then we have that for any positive integer m, Σ
i(m + n,m + p) ⊂
W kℓ+1(m+ n,m+ p).
Proof. Let z = jk0f ∈ Σ
i(m + n,m+ p). Setting α = j10f , we identify α with
the homomorphism Rm+n → Rm+p. Let Ker(α)⊥ and Im(α)⊥ be the orthogonal
complement of the kernel Ker(α) and the image Im(α) of α respectively. Let L
and M be subspaces of Ker(α)⊥ and Im(α) of dimension m such that α maps
L onto M isomorphically. Let L⊥ and M⊥ be their orthogonal complements
in Ker(α)⊥ and Im(α) respectively. Then α is decomposed as in the following
exact sequence.
0 → Ker(α) → L⊕ L⊥ ⊕Ker(α)
→ M ⊕M⊥ ⊕ Im(α)⊥ → Im(α)⊥ → 0
Let us choose coordinates
(u1, · · · , um), (um+1, · · · , um+n−i) and (um+n−i+1, · · · , um+n)
of L, L⊥ and Ker(α), and coordinates
(y1, · · · , ym), (ym+1, · · · , ym+n−i) and (ym+n−i+1, · · · , ym+p)
of M , M⊥ and Im(α)⊥ respectively. Since α maps L onto M isomorphically,
there exist the new coordinates (x1, · · · , xm+n) of R
m+n such that
xj = xj(u1, · · · , um+n) (1 ≤ j ≤ m) and xj = uj (m+ 1 ≤ j ≤ m+ n)
and that
yj ◦ f(x1, · · · , xm+n) = xj (1 ≤ j ≤ m). (4.1)
Setting
x = (xm+1, · · · , xm+n), we define the map g : (R
n, 0) → (Rp, 0) by
yj ◦ g(
x) = yj ◦ f(0, · · · , 0,
x) (m+ 1 ≤ j ≤ m+ p).
Then f is an unfolding of g by (4.1) and g is of kernel rank i at the origin. We
next prove by following the argument and the notation used in [MaIV, Section
1] that d(g,K) is equal to d(f,K). Define π : θ(f) → θ(g) by
ajtf(
j=m+1
j=m+1
◦ g),
where aj ∈ C
∞(Rm+n, 0), a′j ∈ C
∞(Rn, 0) and a′j(
x) = aj(0, · · · , 0,
x). We note
tf(∂/∂xj) = (∂/∂yj) ◦ f +
t=m+1(∂yt ◦ f/∂xj)(∂/∂yt) ◦ f (1 ≤ j ≤ m),
tf(∂/∂xj) =
t=m+1(∂yt ◦ f/∂xj)(∂/∂yt) ◦ f (m+ 1 ≤ j ≤ m+ n),
(∂yt ◦ f/∂xj)(0, · · · , 0,
x) = (∂yt ◦ g/∂xj)(
x) (m+ 1 ≤ t ≤ m+ p).
Since
yt ◦ f(x1, · · · , xm+n)− yt ◦ f(0, · · · , 0,
xubu(x1, · · · , xm+n),
for some bj ∈ C
∞(Rm+n, 0), we have
∂yt ◦ f/∂xj − ∂yt ◦ g/∂xj =
xu(∂bu/∂xj) (m+ 1 ≤ j ≤ m+ n).
Hence, the assertion follows from an elementary calculation under the definition
in (2.3).
Since jk0 g ∈ Σ
i(n, p) ⊂ W kℓ+1(n, p), we have d(g,K) ≧ ℓ+ 1. Hence, we have
d(f,K) ≧ ℓ+ 1. This shows z ∈ W kℓ+1(m+ n,m+ p). This is what we want.
Let ξ be a stable vector bundle over a space. Let c(Σi, ξ) denote the de-
terminant of the (p− n+ i)-matrix whose (s, t)-component is the (i+ s− t)-th
Stiefel-Whitney class Wi+s−t(ξ). If n − p and i are even, say n − p = 2u and
i = 2v, and if ξ is orientable, then cZ(Σ
i, ξ) expresses the determinant of the
(v − u)-matrix whose (s, t)-component is the (v + s − t)-th Pontrjagin class
Pv+s−t(ξ).
Wi · · · Wn−p+1
. . .
Wp−n+2i−1 · · · Wi
Pv · · · Pu+1
. . .
P2v−u−1 · · · Pv
Let τX denote the stable tangent bundle of a manifold X . If f : N → P is a
smooth map transverse to Σi(N,P ) and ξ = τN − f
∗(τP ), then c(Σ
i, ξ) (resp.
i, ξ)) is equal to the (resp. integer) Thom polynomial of the topological
closure of (jkf)−1(Σi(N,P )) ([Po], [Ro] and see also [An1, Proposition 5.4]). If
it does not vanish, then (jkf)−1(Σi(N,P )) cannot be empty by the obstruction
theory in [St]. Hence, we have the following corollary of Propositions 4.2 and
Corollary 4.4 Let f : M → Q be a smooth map with dimM = m + n and
dimQ = m+p. Under the same assumption of Proposition 4.2. we assume that
either
(i) c(Σi, τM − f
∗(τQ)) does not vanish, or
(ii) M and τM −f
∗(τQ) are orientable, n−p and i are even and cZ(Σ
i, τM −
f∗(τQ)) does not vanish.
Then f is not homotopic to any Okℓ -regular map.
5 Homotopy equivalences
In this section we will study the filtration in (1.1) in Introduction by applying
Corollaries 3.7 and 4.4 and Remark 3.8.
Let us first review what is called the Sullivan’s exact sequence in the surgery
theory following [M-M] (see also [K-M], [Su] and [Br]).
In what follows P is a closed and oriented n-manifold. We define the set S(P )
to be the set of all equivalence classes of homotopy equivalences f : N → P of
degree 1 under the following equivalence relation. Let Nj be closed oriented n-
manifolds and let fj : Nj → P be homotopy equivalences of degree 1 (j = 1, 2).
We say that f1 and f2 are equivalent if there exists an h-cobordism W of N1
and N2 and a homotopy equivalence F : (W,N1 ∪ (−N2)) → (P × [0, 1], P × 0∪
(−P )× 1) of degree 1 such that F |Nj = fj (j = 1, 2).
Let O(k) denote the rotation group of Rk and let Gk denote the space of all
homotopy equivalence of the (k − 1)-sphere Sk−1 equipped with the compact-
open topology. By considering the canonical inclusions O(k) → O(k + 1) and
Gk → Gk+1, we set O = limk→∞ O(k) and G = limk→∞ Gk. Let BO and BG
denote the classifying spaces for O and G. Then we have the canonical maps
π(m) : BO(m) → BG(m) and π : BO → BG, which are regarded as fibrations
with fibers G(m)/O(m) and G/O respectively. For a sufficiently large number
m, let ηO(m) denote the universal vector bundle over BO(m) and let iG/O :
G(m)/O(m) → BO(m) be the inclusion of a fiber. Set ηG/O = (iG/O)
∗ηO(m).
Then ηG/O has a trivialization tG/O : ηG/O → R
m as a spherical fibration.
We next recall the surgery obstruction sP4q : [P,G/O] → Z only in the case
of n = 4q. For [α] ∈ [P,G/O] let η = α∗(ηG/O) with the canonical bundle map
α : η → ηG/O covering α and the projection πη onto P . We deform tG/O ◦ α
to a map transverse to 0 ∈ Rm and let M be the inverse image of 0 with a
map πη|M : M → P of degree 1. We define s
4q([α]) = (1/8)(σ(M) − σ(P )).
If P is simply connected in addition, then there have been defined an injection
jP : S(P ) → [P,G/O] such that if sP4q([α]) = 0, πη|M is deformed to a homotopy
equivalence f : N → P of degree 1 under a certain cobordism. The following is
the Sullivan’s exact sequence.
0 −→ S(P )
−→ [P,G/O]
Let us recall the cobordism group Ωh−eqn of homotopy equivalences of degree
1 in [An5]. Let Nj and Pj be oriented closed n-manifolds and let fj : Nj → Pj
be homotopy equivalences of degree 1 (j = 1, 2). We say that f1 and f2 are
cobordant if there exists an oriented (n + 1)-manifold W , V and a homotopy
equivalence F : (W,∂W ) → (V, ∂V ) of degree 1 such that ∂W = N1 ∪ (−N2),
∂V = P1 ∪ (−P2) and F |Nj = fj . The cobordism class of f : N → P is denoted
by [f : N → P ]. Let Ωh−eqn denote the set which consists of all cobordism
classes of homotopy equivalences of degree 1. We provide Ωh−eqn with a module
structure by setting
• [f1 : N1 → P1] + [f2 : N2 → P2] = [f1 ∪ f2 : N1 ∪N2 → P1 ∪ P2],
• −[f : N → P ] = [f : (−N) → (−P )].
The null element is defined to be [f : N → P ] which bound a homotopy equiv-
alence F : (W,∂W ) → (V, ∂V ) of degree 1 such that ∂W = N , ∂V = P and
F |N = f . Even if P is not simply connected, we can find f1 : N1 → P1 with P1
being simply connected in the same cobordism class by killing π1(N) ≈ π1(P )
by usual surgery.
Let cQ(Σ
2i, ηG/O) denote the image of cZ(Σ
2i, ηG/O) in H
4i2(G/O;Q). Let
α = jP ([f : N → P ]) and let cP : P → BSO be a classifying map of the
tangent bundle TP of P . Then it induces the homomorphism C2i : Ω
H4q−4i2(G/O;Q) defined by
C2i([f : N → P ]) = cQ(Σ
2i, ηG/O) ∩ α([P ])
= cQ(Σ
2i, ηG/O)⊗ 1 ∩ (α × cP )∗([P ]),
under the identification
H4q−4i2 (G/O;Q) = H4q−4i2(G/O;Q)⊗ 1
j=0 H4j(G/O;Q)⊗H4q−4i2−4j(BSO;Q). We have that
C2i(α) = cQ(Σ
2i, ηG/O) ∩ (α)∗([P ])
= cQ(Σ
2i, ηG/O) ∩ (α ◦ f)∗([N ])
= (α ◦ f)∗((α ◦ f)
∗(cQ(Σ
2i, ηG/O)) ∩ [N ])
= (α ◦ f)∗(cZ(Σ
2i, τN − f
∗(τP )) ∩ [N ]).
Furthermore, we have proved in [An5, Theorems 3.2 and 4.1] that for integers
q and i with q ≧ i2 ≧ 1,
4q /(Ω
4q ∩Ker(C2i))⊗ Q = dimH4q−4i2 (BSO;Q). (5.1)
The following theorem follows from (5.1), Proposition 4.2 and Corollary 4.4.
Theorem 5.1 Let ℓ, q and i be integers with ℓ ≧ 0 and q ≧ i2. Let k ≧
4q + ℓ + 1. There exists a cobordism class [f : N → P ] ∈ Ω
4q such that
2i, τN − f
∗(τP )) is not a torsion element and that if 4i
3 − 2i2 ≧ 4q + ℓ ≧
4i2 + ℓ, then f is not cobordant in Ω
4q to any O
ℓ -regular map.
We can prove the following theorem using Theorem 5.1 by applying the same
argument in the proof of [An5, Theorem 0.2]. However, Theorem 1.2 is very
important in the following and the situation is rather different. Therefore, we
give its proof.
Theorem 5.2 Let ℓ, q and i be given integers with ℓ ≧ 0 and q ≧ i2. Let
k ≧ 8q + ℓ + 1. If 4i3 − 2i2 ≧ 4q + ℓ ≧ 4i2 + ℓ, then there exists a closed
connected oriented 8q-manifold P and a homotopy equivalence f : P → P of
degree 1 such that cZ(Σ
2i, τP −f
∗(τP )) 6= 0 and that f is not cobordant in Ω
to any Okℓ -regular homotopy equivalence of degree 1.
Proof. It follows from Theorem 5.1 that there exists a homotopy equivalence
f : N → P of degree 1 between 4q-manifolds such that cZ(Σ
2i, τN − f
∗(τP )) is
not a torsion element. Let f−1 : P → N be a homotopy inverse of f . Define
g : N×P → N×P by g(x, y) = (f−1(y), f(x)). We have k ≥ dimN×P + ℓ+1.
If we prove that cZ(Σ
2i, τN×P − g
∗(τN×P )) does not vanish, then, by Corollary
4.4, g is not homotopic to any Okℓ -regular map. We set ξ = τN×P −g
∗(τN×P ) =
τN × τP − f
∗(τP )× (f
−1)∗(τN ). Then
pj(ξ) =
s+t=j
ps(τN × τP )pt(f
∗(τP )× (f
−1)∗(τN ))
s+t=j
s1 + s2 = s
t1 + t2 = t
ps1(τN )pt1(f
∗(τP ))⊗ ps2(τP )pt2((f
−1)∗(τN ))
modulo torsion in H∗(N ;Z) ⊗ H∗(P ;Z). The term of pj(ξ) which lies in
H4j(N ;Z)⊗H0(P ;Z) is equal modulo torsion to
s+t=j
ps(τN )pt(f
∗(τP ))⊗ 1 = pj(τN − f
∗(τP ))⊗ 1.
Hence, we have that cZ(Σ
2i, τN×P−g
∗(τN×P ) is equal to the sum of cZ(Σ
2i, τN−
f∗(τP )) ⊗ 1 and the other term which lies in Σ
4i2−4j(N ;Z) ⊗ H4j(P ;Z)
modulo torsion. Since cZ(Σ
2i, τN − f
∗(τP )) does not vanish, it follows that
2i, τN×P − g
∗(τN×P )) does not vanish. This completes the proof.
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3. In the proof k refers to a sufficiently large integer.
Let i0 = 2, which is the smallest integer such that 4i
3 − 2i2 ≧ 4i2 with q = 4
and ℓ = 8. Then we have, by Theorem 5.2, a closed connected oriented 8 · 4-
manifold P0 and a homotopy equivalence f0 : P0 → P0 of degree 1 such that
4, τP0 − f
0 (τP0)) 6= 0 and that f0 is not homotopic to any O
8 -regular map.
By Remark 3.8 there exists an integer ℓ such that f0 is homotopic to an O
regular map. Let ℓ1 be such a smallest integer.
We assume the following (A-t) for an integer t ≧ 0, where ℓ0 = 8.
(A-t) We have constructed integers ℓt, ℓt+1, it, a closed oriented 8·i
t -manifold
Pt and an O
-regular homotopy equivalence ft : Pt → Pt of degree 1 such
that 4i3t − 2i
t ≧ 4i
t + ℓt, ℓt+1 > ℓt, cZ(Σ
2it , τPt − f
t (τPt)) 6= 0 and that ft is
not homotopic to any Okℓt-regular map.
Under the assumption (A-t) we prove (A-(t+ 1)) with ℓt+1 < ℓt+2. Let it+1
be the smallest integer among the integers i > 0 with 4i3 − 2i2 ≧ 4i2 + ℓt+1.
Then it follows from Theorem 5.2 that there exist a closed connected oriented
8 · i2t+1-manifold Pt+1 and a homotopy equivalence ft+1 : Pt+1 → Pt+1 of degree
1 such that cZ(Σ
2it , τPt+1 − f
t+1(τPt+1)) 6= 0 and that ft+1 is not homotopic to
any Okℓt+1-regular map. It follows Remark 3.8 that there exists an integer ℓ such
that ft+1 is homotopic to an O
ℓ -regular map. Let ℓt+2 be the smallest integer
among those integers ℓ. Hence, we have ℓt+2 > ℓt+1. This proves (A-(t+ 1)).
Thus we have defined the sequences {it}, {ℓt}, closed connected oriented
manifolds {Pt} of dimensions {8 · i
t} and homotopy equivalences {ft} of degree
1 which satisfy the above properties.
Given a positive integer d, let
P = P0 × P1 × P2 × · · · × Pd,
Ft = idP0 × · · · × idPt−1 × ft × idPt+1 × · · · × idPd (0 ≦ t ≦ d),
and p =
t=0 8 · i
t . We show that Ft /∈ hℓt(P ) and Ft ∈ hℓt+1(P ). Let
qt : P → Pt be the canonical projection. Then the stable tangent bundle τP is
isomorphic to q∗0(τP0)⊕ q
1(τP1 )⊕ · · · ⊕ q
d(τPd). Hence, τP − F
t (τP ) is equal to
q∗0(τP0)⊕ q
1(τP1)⊕ · · · ⊕ q
d(τPd)
− ((q0 ◦ Ft)
∗(τP0 )⊕ (q1 ◦ Ft)
∗(τP1)⊕ · · · ⊕ (qd ◦ Ft)
∗(τPd))
= q∗0(τP0 )⊕ q
1(τP1)⊕ · · · ⊕ q
d(τPd)
− (q∗0(τP0 )⊕ · · · ⊕ q
t−1(τPt−1 )⊕ (ft ◦ qt)
∗(τPt)⊕ · · · ⊕ q
d(τPd))
= q∗t (τPt)− (ft ◦ qt)
∗(τPt)
= q∗t ((τPt)− f
t (τPt)).
This shows that
2it , τP − F
t (τP )) = cZ(Σ
2it , q∗t ((τPt)− f
t (τPt))
= q∗t (cZ(Σ
2it , τPt − f
t (τPt)),
which does not vanish in H2i
t (P ;Z) since cZ(Σ
2it , τPt − f
t (τPt)) 6= 0 and since
q∗t : H
t (Pt;Z) → H
t (P ;Z) is injective. Furthermore, it follows from Propo-
sition 4.3 that Σ2it(p, p) ⊂ W kℓ+1(p, p) and from Corollary 4.4 that Ft is not
homotopic to any Okℓt -regular map. However, since ft is homotopic to an O
regular map, Ft is also homotopic to an O
-regular map. This proves the
theorem.
We prepare further results which are necessary to study the filtration in
(1.1). The assertions (i) and (ii) in the following theorem have been proved in
[An2, Theorem 4.8] and [An4, Theorem 4.1] respectively, which are applications
of the relative homotopy principles for O-regular maps.
Theorem 5.3 Let P be orientable and f : P → P be a smooth map.
(i) A map f is homotopic to a fold-map if and only if τP is isomorphic to
f∗(τP ).
(ii) If a map f is Ω1-regular, then f is homotopic to an Ω(1,1,0)-regular map.
Let V (n, p) be an algebraic set of Jk(n, p) which is invariant with respect
to the actions of local diffeomorphisms of (Rn, 0) and (Rn, 0) and Let V (N,P )
be the subbundle of Jk(N,P ) associated to V (n, p). By [B-H] we have the
fundamental class of V (N,P ) under the coefficient group Z/2, and have the
Thom polynomial c(V (n, p), τN − f
∗(τP )) of V (N,P ).
Theorem 5.4 Let V (p, p) be as above. Let P be orientable and f : P → P be
a smooth map.
(i) If f is a homotopy equivalence, then c(V (p, p), τP − f
∗(τP )) vanishes.
(ii) cZ(W
p (p, p), τP − f
∗(τP )) = 0 for p = 5, 6, 7 and
8 (8, 8), τP − f
∗(τP )) = 9P2(τP − f
∗(τP )) + 3P
1 (τP − f
∗(τP ))
for p = 8.
(iii) Let 2 ≦ p ≦ 8. Then there exists a section s of Okp−1(P, P ) over P with
πkP ◦ s and f being homotopic if and only if cZ(W
p (p, p), τP − f
∗(τP )) = 0.
Proof. (i) Let S(νP ) denote the spherical normal fiber space of P . It follows
from [Sp] that S(νP ) is equivalent to f
∗(S(νP )). Hence, the associated spherical
spaces of τP and f
∗(τP ) are equivalent. In particular, the Stiefel-Whitney classes
of τP − f
∗(τP ) vanish.
(ii) If p ≦ 8, then a map f : P → P is homotopic to a smooth map with
only K-simple singularities by [MaVI]. According to [F-R], the integer Thom
polynomial ofW kp (p, p) is equal to the formula for p = 8 and vanish for p = 5, 6, 7
in Hp(P ;Z) ≈ Z.
(iii) It follows from the relative homotopy principle for Okp−1-regular maps
P → P that the primary obstruction in Hp(P ;πp−1(O
p−1(p, p)) is the unique
obstruction for finding the required section. By an elementary argument we
πp−1(O
p−1(p, p)) ≈ Hp−1(O
p−1(p, p);Z) ≈ H
dimWk
(p,p)(W kp (p, p);Z).
This shows the assertion.
Finally we study the filtration in (1.1) in the case of P being orientable and
p ≦ 8 by applying the homotopy principles in Theorems 1.2 and 5.3. We have
hp(P ) = h(P ).
Examples.
Case: p ≦ 3; h0(P ) ⊂ h1(P ) = h(P ).
Since P is parallelizable, TP and f∗(TP ) are trivial. So a map f : P → P
is homotopic to a fold-map. We refer the reader to [Ru, 1].
Case: p = 4; h0(P ) ⊂ h1(P ) ⊂ h2(P ) = h3(P ) ⊂ h4(P ).
It is known that cZ(Σ
4; τP − f
∗(τP )) = P2(τP − f
∗(τP )). If this class van-
ish, then there exists a section P → Ω1(P, P ) covering f , and hence an Ω1-
regular map by [F]. By Theorems 5.3 and 5.4 we obtain an Ω(1,1,0)-regular
map homotopic to f . It has been proved in [Ak] that h0(P ) 6= h(P ) for
P = S3 × S1#S2 × S2.
Case: 5 ≦ p ≦ 7; h0(P ) ⊂ h1(P ) ⊂ · · · ⊂ hp−1(P ) = hp(P ).
This follows from Theorems 1.2 and 5.4.
Case: p = 8; h0(P ) ⊂ h1(P ) ⊂ · · · ⊂ h7(P ) ⊂ h8(P ).
If 9P2(τP − f
∗(τP )) + 3P
1 (τP − f
∗(τP )) = 0, then the homotopy class of f
lies in h7(P ) by Theorems 1.2 and 5.4.
For more precise information we must investigate the obstructions for finding
sections in Γ
(P, P ) related to W kℓ+1(p, p).
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Department of Mathematical Sciences
Faculty of Science, Yamaguchi University
Yamaguchi 753-8512, Japan
E-mail: andoy@yamaguchi-u.ac.jp
	Introduction
	Boardman manifolds and K-orbits
	Proofs of Theorems 1.1 and 1.2.
	Nonexistence theorems
	Homotopy equivalences
