Cobordism

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 ${\displaystyle \phi :{\mathbb{S}}^p\times {\mathbb{D}}^q\subset M,}$ define the n-manifold

${\displaystyle N:=(M-\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{m}\phi ){\cup }_{\phi {|}_{{\mathbb{S}}^p\times {\mathbb{S}}^{q-1}}}\left({\mathbb{D}}^{p+1}\times {\mathbb{S}}^{q-1}\right)}$

obtained by surgery, via cutting out the interior of ${\displaystyle {\mathbb{S}}^p\times {\mathbb{D}}^q}$ and gluing in ${\displaystyle {\mathbb{D}}^{p+1}\times {\mathbb{S}}^{q-1}}$ along their boundary

${\displaystyle \mathrm{\partial }\left({\mathbb{S}}^p\times {\mathbb{D}}^q\right)={\mathbb{S}}^p\times {\mathbb{S}}^{q-1}=\mathrm{\partial }\left({\mathbb{D}}^{p+1}\times {\mathbb{S}}^{q-1}\right).}$

The trace of the surgery

${\displaystyle W:=(M\times I){\cup }_{{\mathbb{S}}^p\times {\mathbb{D}}^q\times \{1\}}\left({\mathbb{D}}^{p+1}\times {\mathbb{D}}^q\right)}$

defines an elementary cobordism (W; M, N). Note that M is obtained from N by surgery on ${\displaystyle {\mathbb{D}}^{p+1}\times {\mathbb{S}}^{q-1}\subset N.}$ 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 ${\displaystyle {\mathbb{S}}^0\times {\mathbb{D}}^1}$ and gluing in ${\displaystyle {\mathbb{D}}^1\times {\mathbb{S}}^0.}$ The pictures in Fig. 1 show that the result of doing this is either (i) ${\displaystyle {\mathbb{S}}^1}$ again, or (ii) two copies of ${\displaystyle {\mathbb{S}}^1}$

For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either ${\displaystyle {\mathbb{S}}^0\times {\mathbb{D}}^2}$ or ${\displaystyle {\mathbb{S}}^1\times {\mathbb{D}}^1.}$

1. ${\displaystyle {\mathbb{S}}^1\times {\mathbb{D}}^1}$: If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in ${\displaystyle {\mathbb{S}}^0\times {\mathbb{D}}^2}$ – that is, two disks - and it's clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)
2. ${\displaystyle {\mathbb{S}}^0\times {\mathbb{D}}^2}$: Having cut out two disks ${\displaystyle {\mathbb{S}}^0\times {\mathbb{D}}^2,}$ we glue back in the cylinder ${\displaystyle {\mathbb{S}}^1\times {\mathbb{D}}^1.}$ There are two possible outcomes, depending on whether our gluing maps have the same or opposite orientation on the two boundary circles. If the orientations are the same (Fig. 2b), the resulting manifold is the torus ${\displaystyle {\mathbb{S}}^1\times {\mathbb{S}}^1}$ but if they are different, we obtain the Klein bottle (Fig. 2c).

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 (Dieudonné 1989, p. 289). 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 ${\displaystyle {\mathfrak{N}}_n}$ (rather than the more systematic ${\displaystyle {\mathrm{\Omega }}_n^{\text{O}}}$); 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 ${\displaystyle [M]+[N]=[M\bigsqcup N]}$; this is a well-defined operation which turns ${\displaystyle {\mathfrak{N}}_n}$ into an abelian group. The identity element of this group is the class ${\displaystyle [\mathrm{\varnothing }]}$ consisting of all closed n-manifolds which are boundaries. Further we have ${\displaystyle [M]+[M]=[\mathrm{\varnothing }]}$ for every M since ${\displaystyle M\bigsqcup M=\mathrm{\partial }(M\times [0,1])}$. Therefore, ${\displaystyle {\mathfrak{N}}_n}$ is a vector space over ${\displaystyle {\mathbb{F}}_2}$, the field with two elements. The cartesian product of manifolds defines a multiplication ${\displaystyle [M][N]=[M\times N],}$ so

${\displaystyle {\mathfrak{N}}_{\ast }=\underset{n⩾0}{⨁}{\mathfrak{N}}_n}$

is a graded algebra, with the grading given by the dimension.

The cobordism class ${\displaystyle [M]\in {\mathfrak{N}}_n}$ 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 ${\displaystyle [M]=0\in {\mathfrak{N}}_n}$. In 1954 René Thom proved

${\displaystyle {\mathfrak{N}}_{\ast }={\mathbb{F}}_2\left[x_i|i⩾1,i\ne 2^j-1\right]}$

the polynomial algebra with one generator ${\displaystyle x_i}$ in each dimension ${\displaystyle i\ne 2^j-1}$. Thus two unoriented closed n-dimensional manifolds M, N are cobordant, ${\displaystyle [M]=[N]\in {\mathfrak{N}}_n,}$ if and only if for each collection ${\displaystyle \left(i_1,\cdots ,i_k\right)}$ of k-tuples of integers ${\displaystyle i⩾1,i\ne 2^j-1}$ such that ${\displaystyle i_1+\cdots +i_k=n}$ the Stiefel-Whitney numbers are equal

${\displaystyle ⟨w_{i_1}(M)\cdots w_{i_k}(M),[M]⟩=⟨w_{i_1}(N)\cdots w_{i_k}(N),[N]⟩\in {\mathbb{F}}_2}$

with ${\displaystyle w_i(M)\in H^i\left(M;{\mathbb{F}}_2\right)}$ the ith Stiefel-Whitney class and ${\displaystyle [M]\in H_n\left(M;{\mathbb{F}}_2\right)}$ the ${\displaystyle {\mathbb{F}}_2}$-coefficient fundamental class.

For even i it is possible to choose ${\displaystyle x_i=\left[{\mathbb{P}}^i(\mathbb{R})\right]}$, the cobordism class of the i-dimensional real projective space.

The low-dimensional unoriented cobordism groups are

${\displaystyle \begin{array}{cc}{\mathfrak{N}}_0& =\mathbb{Z}/2,\\ {\mathfrak{N}}_1& =0,\\ {\mathfrak{N}}_2& =\mathbb{Z}/2,\\ {\mathfrak{N}}_3& =0,\\ {\mathfrak{N}}_4& =\mathbb{Z}/2\oplus \mathbb{Z}/2,\\ {\mathfrak{N}}_5& =\mathbb{Z}/2.\end{array}}$

This shows, for example, that every 3-dimensional closed manifold is the boundary of a 4-manifold (with boundary).

The Euler characteristic ${\displaystyle \chi (M)\in \mathbb{Z}}$ modulo 2 of an unoriented manifold M is an unoriented cobordism invariant. This is implied by the equation

${\displaystyle {\chi }_{\mathrm{\partial }W}=\left(1-(-1{)}^{\dim W}\right){\chi }_W}$

for any compact manifold with boundary ${\displaystyle W}$.

Therefore, ${\displaystyle \chi :{\mathfrak{N}}_i\to \mathbb{Z}/2}$ is a well-defined group homomorphism. For example, for any ${\displaystyle i_1,\cdots ,i_k\in \mathbb{N}}$

${\displaystyle \chi \left({\mathbb{P}}^{2i_1}(\mathbb{R})\times \cdots \times {\mathbb{P}}^{2i_k}(\mathbb{R})\right)=1.}$

In particular such a product of real projective spaces is not null-cobordant. The mod 2 Euler characteristic map ${\displaystyle \chi :{\mathfrak{N}}_{2i}\to \mathbb{Z}/2}$ is onto for all ${\displaystyle i\in \mathbb{N},}$ and a group isomorphism for ${\displaystyle i=1.}$

Moreover, because of ${\displaystyle \chi (M\times N)=\chi (M)\chi (N)}$, these group homomorphisms assemble into a homomorphism of graded algebras:

${\displaystyle \{\begin{array}{l}\mathfrak{N}\to {\mathbb{F}}_2[x]\\ [M]\mapsto \chi (M)x^{\dim (M)}\end{array}}$

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 ${\displaystyle {\mathbb{R}}^{n+k}}$ 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 X_k → X_k+1 with maps X_k → BO(k) (compatible with the inclusions BO(k) → BO(k+1), an X-structure is a lift of ν to a map ${\displaystyle \tilde{\nu }:M\to X_k}$. Considering only manifolds and cobordisms with X-structure gives rise to a more general notion of cobordism. In particular, X_k 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 ${\displaystyle {\mathrm{\Omega }}_{\ast }^G}$.

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 ${\displaystyle M\bigsqcup (-N)}$, where −N denotes N with the reversed orientation. For example, boundary of the cylinder M × I is ${\displaystyle M\bigsqcup (-M)}$: 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, 2M is not in general an oriented boundary, that is, 2[M] ≠ 0 when considered in ${\displaystyle {\mathrm{\Omega }}_{\ast }^{\text{SO}}.}$

The oriented cobordism groups are given modulo torsion by

${\displaystyle {\mathrm{\Omega }}_{\ast }^{\text{SO}}\otimes \mathbb{Q}=\mathbb{Q}\left[y_{4i}\mid i⩾1\right],}$

the polynomial algebra generated by the oriented cobordism classes

${\displaystyle y_{4i}=\left[{\mathbb{P}}^{2i}(\mathbb{C})\right]\in {\mathrm{\Omega }}_{4i}^{\text{SO}}}$

of the complex projective spaces (Thom, 1952). The oriented cobordism group ${\displaystyle {\mathrm{\Omega }}_{\ast }^{\text{SO}}}$ 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 :

${\displaystyle \begin{array}{cc}{\mathrm{\Omega }}_0^{\text{SO}}& =\mathbb{Z},\\ {\mathrm{\Omega }}_1^{\text{SO}}& =0,\\ {\mathrm{\Omega }}_2^{\text{SO}}& =0,\\ {\mathrm{\Omega }}_3^{\text{SO}}& =0,\\ {\mathrm{\Omega }}_4^{\text{SO}}& =\mathbb{Z},\\ {\mathrm{\Omega }}_5^{\text{SO}}& ={\mathbb{Z}}_2.\end{array}}$

The signature of an oriented 4i-dimensional manifold M is defined as the signature of the intersection form on ${\displaystyle H^{2i}(M)\in \mathbb{Z}}$ and is denoted by ${\displaystyle \sigma (M).}$ It is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the Hirzebruch signature theorem.

For example, for any i₁, ..., i_k ≥ 1

${\displaystyle \sigma \left({\mathbb{P}}^{2i_1}(\mathbb{C})\times \cdots \times {\mathbb{P}}^{2i_k}(\mathbb{C})\right)=1.}$

The signature map ${\displaystyle \sigma :{\mathrm{\Omega }}_{4i}^{\text{SO}}\to \mathbb{Z}}$ 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 ${\displaystyle {\mathrm{\Omega }}_n^G(X)}$ and cohomology ("cobordism") groups ${\displaystyle {\mathrm{\Omega }}_G^n(X)}$ for any space X. The generalized homology groups ${\displaystyle {\mathrm{\Omega }}_{\ast }^G(X)}$ are covariant in X, and the generalized cohomology groups ${\displaystyle {\mathrm{\Omega }}_G^{\ast }(X)}$ are contravariant in X. The cobordism groups defined above are, from this point of view, the homology groups of a point: ${\displaystyle {\mathrm{\Omega }}_n^G={\mathrm{\Omega }}_n^G(\text{pt})}$. Then ${\displaystyle {\mathrm{\Omega }}_n^G(X)}$ 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] ∈ H_n(M) (with coefficients in ${\displaystyle \mathbb{Z}/2}$ in general, and in ${\displaystyle \mathbb{Z}}$ in the oriented case), defining a natural transformation

${\displaystyle \{\begin{array}{l}{\mathrm{\Omega }}_n^G(X)\to H_n(X)\\ (M,f)\mapsto f_{\ast }[M]\end{array}}$

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 ${\displaystyle {\mathrm{\Omega }}_G^n(X)}$ 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

${\displaystyle {\mathrm{\Omega }}_n^G(X)=\sum _{p+q=n}{H}_p(X;{\mathrm{\Omega }}_q^G(\text{pt})).}$

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 MG_n of the standard vector bundles over the classifying spaces BG_n. 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(π_∗(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.

Other results

In 1959, C.T.C. Wall proved that two manifolds are cobordant if and only if their Pontrjagin numbers and Stiefel numbers are the same.