Cobordism

In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French "bord", giving "cobordism") of a manifold. Two manifolds of the same dimension are "cobordant" if their disjoint union is the "boundary" of a compact manifold one dimension higher.
The boundary of an ("n" + 1)-dimensional manifold "W" is an "n"-dimensional manifold ∂"W" that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for
piecewise linear and topological manifolds.
A "cobordism" between manifolds "M" and "N" is a compact manifold "W" whose boundary is the disjoint union of "M" and "N", formula_1.
Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected with Morse theory, and "h"-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory. In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.
Definition.
Manifolds.
Roughly speaking, an "n"-dimensional manifold "M" is a topological space locally (i.e., near each point) homeomorphic to an open subset of Euclidean space formula_2 A manifold with boundary is similar, except that a point of "M" is allowed to have a neighborhood that is homeomorphic to an open subset of the half-space
Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of formula_4; the boundary of formula_4 is denoted by formula_6. Finally, a closed manifold is, by definition, a compact manifold without boundary (formula_7.)
Cobordisms.
An formula_8-dimensional "cobordism" is a quintuple formula_9 consisting of an formula_8-dimensional compact differentiable manifold with boundary, formula_11; closed formula_12-manifolds formula_4, formula_14; and embeddings formula_15, formula_16 with disjoint images such that
The terminology is usually abbreviated to formula_18. "M" and "N" are called "cobordant" if such a cobordism exists. All manifolds cobordant to a fixed given manifold "M" form the "cobordism class" of "M".
Every closed manifold "M" is the boundary of the non-compact manifold "M" × [0, 1); for this reason we require "W" to be compact in the definition of cobordism. Note however that "W" is "not" required to be connected; as a consequence, if "M" = ∂"W"1 and "N" = ∂"W"2, then "M" and "N" are cobordant.
Examples.
The simplest example of a cobordism is the unit interval "I" = [0, 1]. It is a 1-dimensional cobordism between the 0-dimensional manifolds {0}, {1}. More generally, for any closed manifold "M", ("M" × "I"; "M" x {0}, "M" x {1}) is a cobordism from "M" × {0} to "M" × {1}.
If "M" consists of a circle, and "N" of two circles, "M" and "N" together make up the boundary of a pair of pants "W" (see the figure at right). Thus the pair of pants is a cobordism between "M" and "N". A simpler cobordism between "M" and "N" is given by the disjoint union of three disks.
The pair of pants is an example of a more general cobordism: for any two "n"-dimensional manifolds "M", "M"′, the disjoint union formula_19 is cobordant to the connected sum formula_20 The previous example is a particular case, since the connected sum formula_21 is isomorphic to formula_22 The connected sum formula_23 is obtained from the disjoint union formula_19 by surgery on an embedding of formula_25 in formula_19, and the cobordism is the trace of the surgery.
Terminology.
An "n"-manifold "M" is called "null-cobordant" if there is a cobordism between "M" and the empty manifold; in other words, if "M" is the entire boundary of some ("n" + 1)-manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a "n"-sphere is null-cobordant since it bounds a ("n" + 1)-disk. Also, every orientable surface is null-cobordant, because it is the boundary of a handlebody. On the other hand, the 2"n"-dimensional real projective space formula_27 is a (compact) closed manifold that is not the boundary of a manifold, as is explained below.
The general "bordism problem" is to calculate the cobordism classes of manifolds subject to various conditions.
Null-cobordisms with additional structure are called fillings. "Bordism" and "cobordism" are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question "bordism of manifolds", and the study of cobordisms as objects "cobordisms of manifolds".
The term "bordism" comes from French , meaning boundary. Hence bordism is the study of boundaries. "Cobordism" means "jointly bound", so "M" and "N" are cobordant if they jointly bound a manifold, i.e., if their disjoint union is a boundary. Further, cobordism groups form an extraordinary "cohomology theory", hence the co-.
Variants.
The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are oriented, or carry some other additional structure referred to as G-structure. This gives rise to "oriented cobordism" and "cobordism with G-structure", respectively. Under favourable technical conditions these form a graded ring called the cobordism ring formula_28, with grading by dimension, addition by disjoint union and multiplication by cartesian product. The cobordism groups formula_28 are the coefficient groups of a generalised homology theory.
When there is additional structure, the notion of cobordism must be formulated more precisely: a "G"-structure on "W" restricts to a "G"-structure on "M" and "N". The basic examples are "G" = O for unoriented cobordism, "G" = SO for oriented cobordism, and "G" = U for complex cobordism using "stably" complex manifolds. Many more are detailed by Robert E. Stong.
In a similar vein, a standard tool in surgery theory is surgery on normal maps: such a process changes a normal map to another normal map within the same bordism class.
Instead of considering additional structure, it is also possible to take into account various notions of manifold, especially piecewise linear (PL) and topological manifolds. This gives rise to bordism groups formula_30, which are harder to compute than the differentiable variants.
Surgery construction.
Recall that in general, if "X", "Y" are manifolds with boundary, then the boundary of the product manifold is ∂("X" × "Y") = (∂"X" × "Y") ∪ ("X" × ∂"Y").
Now, given a manifold "M" of dimension "n" = "p" + "q" and an embedding formula_31 define the "n"-manifold
obtained by surgery, via cutting out the interior of formula_33 and gluing in formula_34 along their boundary
The trace of the surgery
defines an elementary cobordism ("W"; "M", "N"). Note that "M" is obtained from "N" by surgery on formula_37 This is called reversing the surgery.
Every cobordism is a union of elementary cobordisms, by the work of Marston Morse, René Thom and John Milnor.
Examples.
As per the above definition, a surgery on the circle consists of cutting out a copy of formula_38 and glueing in formula_39 The pictures in Fig. 1 show that the result of doing this is either (i) formula_40 again, or (ii) two copies of formula_40
For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either formula_42 or formula_43
Morse functions.
Suppose that "f" is a Morse function on an ("n" + 1)-dimensional manifold, and suppose that "c" is a critical value with exactly one critical point in its preimage. If the index of this critical point is "p" + 1, then the level-set "N" := "f"−1("c" + ε) is obtained from "M" := "f"−1("c" − ε) by a "p"-surgery. The inverse image "W" := "f"−1(["c" − ε, "c" + ε]) defines a cobordism ("W"; "M", "N") that can be identified with the trace of this surgery.
Geometry, and the connection with Morse theory and handlebodies.
Given a cobordism ("W"; "M", "N") there exists a smooth function "f" : "W" → [0, 1] such that "f"−1(0) = "M", "f"−1(1) = "N". By general position, one can assume "f" is Morse and such that all critical points occur in the interior of "W". In this setting "f" is called a Morse function on a cobordism. The cobordism ("W"; "M", "N") is a union of the traces of a sequence of surgeries on "M", one for each critical point of "f". The manifold "W" is obtained from "M" × [0, 1] by attaching one handle for each critical point of "f".
The Morse/Smale theorem states that for a Morse function on a cobordism, the flowlines of "f"′ give rise to a handle presentation of the triple ("W"; "M", "N"). Conversely, given a handle decomposition of a cobordism, it comes from a suitable Morse function. In a suitably normalized setting this process gives a correspondence between handle decompositions and Morse functions on a cobordism.
History.
Cobordism had its roots in the (failed) attempt by Henri Poincaré in 1895 to define homology purely in terms of manifolds . Poincaré simultaneously defined both homology and cobordism, which are not the same, in general. See Cobordism as an extraordinary cohomology theory for the relationship between bordism and homology.
Bordism was explicitly introduced by Lev Pontryagin in geometric work on manifolds. It came to prominence when René Thom showed that cobordism groups could be computed by means of homotopy theory, via the Thom complex construction. Cobordism theory became part of the apparatus of extraordinary cohomology theory, alongside K-theory. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the Hirzebruch–Riemann–Roch theorem, and in the first proofs of the Atiyah–Singer index theorem.
In the 1980s the category with compact manifolds as objects and cobordisms between these as morphisms played a basic role in the Atiyah–Segal axioms for topological quantum field theory, which is an important part of quantum topology.
Categorical aspects.
Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a category whose objects are closed manifolds and whose morphisms are cobordisms. Roughly speaking, composition is given by gluing together cobordisms end-to-end: the composition of ("W"; "M", "N") and ("W" ′; "N", "P") is defined by gluing the right end of the first to the left end of the second, yielding ("W" ′ ∪"N" "W"; "M", "P"). A cobordism is a kind of cospan: "M" → "W" ← "N". The category is a dagger compact category.
A topological quantum field theory is a monoidal functor from a category of cobordisms to a category of vector spaces. That is, it is a functor whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds.
In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not. For instance, the disk bounding the circle corresponds to a nullary (0-ary) operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
Unoriented cobordism.
The set of cobordism classes of closed unoriented "n"-dimensional manifolds is usually denoted by formula_50 (rather than the more systematic formula_51); it is an abelian group with the disjoint union as operation. More specifically, if ["M"] and ["N"] denote the cobordism classes of the manifolds "M" and "N" respectively, we define formula_52; this is a well-defined operation which turns formula_50 into an abelian group. The identity element of this group is the class formula_54 consisting of all closed "n"-manifolds which are boundaries. Further we have formula_55 for every "M" since formula_56. Therefore, formula_50 is a vector space over formula_58, the field with two elements. The cartesian product of manifolds defines a multiplication formula_59 so
is a graded algebra, with the grading given by the dimension.
The cobordism class formula_61 of a closed unoriented "n"-dimensional manifold "M" is determined by the Stiefel–Whitney characteristic numbers of "M", which depend on the stable isomorphism class of the tangent bundle. Thus if "M" has a stably trivial tangent bundle then formula_62. In 1954 René Thom proved
the polynomial algebra with one generator formula_64 in each dimension formula_65. Thus two unoriented closed "n"-dimensional manifolds "M", "N" are cobordant, formula_66 if and only if for each collection formula_67 of "k"-tuples of integers formula_68 such that formula_69 the Stiefel-Whitney numbers are equal
with formula_71 the "i"th Stiefel-Whitney class and formula_72 the formula_58-coefficient fundamental class.
For even "i" it is possible to choose formula_74, the cobordism class of the "i"-dimensional real projective space.
The low-dimensional unoriented cobordism groups are
This shows, for example, that every 3-dimensional closed manifold is the boundary of a 4-manifold (with boundary).
The Euler characteristic formula_76 modulo 2 of an unoriented manifold "M" is an unoriented cobordism invariant. This is implied by the equation
for any compact manifold with boundary formula_11.
Therefore, formula_79 is a well-defined group homomorphism. For example, for any formula_80
In particular such a product of real projective spaces is not null-cobordant. The mod 2 Euler characteristic map formula_82 is onto for all formula_83 and a group isomorphism for formula_84
Moreover, because of formula_85, these group homomorphism assemble into a homomorphism of graded algebras:
Cobordism of manifolds with additional structure.
Cobordism can also be defined for manifolds that have additional structure, notably an orientation. This is made formal in a general way using the notion of "X"-structure (or G-structure). Very briefly, the normal bundle ν of an immersion of "M" into a sufficiently high-dimensional Euclidean space formula_87 gives rise to a map from "M" to the Grassmannian, which in turn is a subspace of the classifying space of the orthogonal group: ν: "M" → Gr("n", "n" + "k") → "BO"("k"). Given a collection of spaces and maps "Xk" → "Xk"+1 with maps "Xk" → "BO"("k") (compatible with the inclusions "BO"("k") → "BO"("k"+1), an "X"-structure is a lift of ν to a map formula_88. Considering only manifolds and cobordisms with "X"-structure gives rise to a more general notion of cobordism. In particular, "Xk" may be given by "BG"("k"), where "G"("k") → "O"("k") is some group homomorphism. This is referred to as a G-structure. Examples include "G" = "O", the orthogonal group, giving back the unoriented cobordism, but also the subgroup SO("k"), giving rise to oriented cobordism, the spin group, the unitary group "U"("k"), and the trivial group, giving rise to framed cobordism.
The resulting cobordism groups are then defined analogously to the unoriented case. They are denoted by formula_28.
Oriented cobordism.
Oriented cobordism is the one of manifolds with an SO-structure. Equivalently, all manifolds need to be oriented and cobordisms ("W", "M", "N") (also referred to as "oriented cobordisms" for clarity) are such that the boundary (with the induced orientations) is formula_90, where −"N" denotes "N" with the reversed orientation. For example, boundary of the cylinder "M" × "I" is formula_91: both ends have opposite orientations. It is also the correct definition in the sense of extraordinary cohomology theory.
Unlike in the unoriented cobordism group, where every element is two-torsion, 2"M" is not in general an oriented boundary, that is, 2["M"] ≠ 0 when considered in formula_92
The oriented cobordism groups are given modulo torsion by
the polynomial algebra generated by the oriented cobordism classes
of the complex projective spaces (Thom, 1952). The oriented cobordism group formula_95 is determined by the Stiefel–Whitney and Pontrjagin characteristic numbers (Wall, 1960). Two oriented manifolds are oriented cobordant if and only if their Stiefel–Whitney and Pontrjagin numbers are the same.
The low-dimensional oriented cobordism groups are :
The signature of an oriented 4"i"-dimensional manifold "M" is defined as the signature of the intersection form on formula_97 and is denoted by formula_98 It is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the Hirzebruch signature theorem.
For example, for any "i"1, ..., "ik" ≥ 1
The signature map formula_100 is onto for all "i" ≥ 1, and an isomorphism for "i" = 1.
Cobordism as an extraordinary cohomology theory.
Every vector bundle theory (real, complex etc.) has an extraordinary cohomology theory called K-theory. Similarly, every cobordism theory Ω"G" has an extraordinary cohomology theory, with homology ("bordism") groups formula_101 and cohomology ("cobordism") groups formula_102 for any space "X". The generalized homology groups formula_103 are covariant in "X", and the generalized cohomology groups formula_104 are contravariant in "X". The cobordism groups defined above are, from this point of view, the homology groups of a point: formula_105. Then formula_101 is the group of "bordism" classes of pairs ("M", "f") with "M" a closed "n"-dimensional manifold "M" (with G-structure) and "f" : "M" → "X" a map. Such pairs ("M", "f"), ("N", "g") are "bordant" if there exists a G-cobordism ("W"; "M", "N") with a map "h" : "W" → "X", which restricts to "f" on "M", and to "g" on "N".
An "n"-dimensional manifold "M" has a fundamental homology class ["M"] ∈ "Hn"("M") (with coefficients in formula_107 in general, and in formula_108 in the oriented case), defining a natural transformation
which is far from being an isomorphism in general.
The bordism and cobordism theories of a space satisfy the Eilenberg–Steenrod axioms apart from the dimension axiom. This does not mean that the groups formula_102 can be effectively computed once one knows the cobordism theory of a point and the homology of the space "X", though the Atiyah–Hirzebruch spectral sequence gives a starting point for calculations. The computation is only easy if the particular cobordism theory reduces to a product of ordinary homology theories, in which case the bordism groups are the ordinary homology groups
This is true for unoriented cobordism. Other cobordism theories do not reduce to ordinary homology in this way, notably framed cobordism, oriented cobordism and complex cobordism. The last-named theory in particular is much used by algebraic topologists as a computational tool (e.g., for the homotopy groups of spheres).
Cobordism theories are represented by Thom spectra "MG": given a group "G", the Thom spectrum is composed from the Thom spaces "MGn" of the standard vector bundles over the classifying spaces "BGn". Note that even for similar groups, Thom spectra can be very different: "MSO" and "MO" are very different, reflecting the difference between oriented and unoriented cobordism.
From the point of view of spectra, unoriented cobordism is a product of Eilenberg–MacLane spectra – "MO" = "H"(∗("MO")) – while oriented cobordism is a product of Eilenberg–MacLane spectra rationally, and at 2, but not at odd primes: the oriented cobordism spectrum "MSO" is rather more complicated than "MO".