Cobordism

A cobordism (W;M,N).

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 of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes from the French word bord for boundary. The boundary of an n + 1-dimensional manifold W is an n-dimensional manifold $\partial W$ 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 for smooth (i.e., differentiable) manifolds, but there are now also versions for piecewise-linear and topological manifolds.

A cobordism is a manifold W with boundary whose boundary is partitioned in two, $\partial W=M \sqcup N$.

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 $\geq 4$ – 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

More formally, an n + 1-dimensional cobordism is a quintuple (W;M,N,i,j) consisting of an (n + 1)-manifold with boundary $\partial W$, closed n-manifolds M,N and embeddings $i:M \subset \partial W$, $j:N \subset \partial W$ with disjoint images such that

$\partial W ~=~ i(M) \sqcup j(N)~.$

The terminology is usually abbreviated to (W;M,N).^[1]

Every closed manifold M is the boundary of the non-compact manifold $M \times [0,1)$. So for the purposes of cobordism theory only compact manifolds are considered.

The above is the most basic (unoriented) form of the definition. In most practical situations, the manifolds in question are oriented, or carry some additional G-structure. One defines "oriented cobordism" and "cobordism with G-structure", but there are various technicalities. In each particular case, cobordism is an equivalence relation on manifolds. A basic question is to determine the equivalence classes for this relationship, called the cobordism classes of manifolds. Under favourable technical conditions these form a graded ring called the cobordism ring $\Omega^G_*$, with grading by dimension, addition by disjoint union and multiplication by cartesian product. The cobordism groups $\Omega^G_*$ are the coefficient groups of a generalised homology theory.

Main ideas

This section sets out some of the mathematical background to cobordism.

Manifolds with boundary

An n+1-dimensional manifold with boundary is a topological space that is a union of the interior and the boundary: each point in the interior has a neighbourhood homeomorphic to the Euclidean n+1-dimensional space R^{n + 1} and each point in the boundary has a neighbourhood homeomorphic to the Euclidean n+1-dimensional half-space $\{(x_1,x_2,...,x_{n+1}) \in R^{n+1} \vert x_{n+1} \geq 0\}$. In both cases, the point corresponds to the origin (0,0,...,0).

Additional geometric structure

In most applications, the manifold M comes with additional geometric structure, such as a map to some reference space, or additional bundle data. One then wants the surgery process to endow M' with the same kind of additional structure. For instance, 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.

The surgery construction of cobordisms

Recall that in general, if X,Y are manifolds with boundary, then the boundary of the product manifold is $\partial(X \times Y)=(\partial X \times Y)\cup(X \times \partial Y)$.

Now, given a manifold $~M$ of dimension n = p + q and an embedding $\phi\colon S^p\times D^q\subset M$, define the n-manifold

$N := (M-\operatorname{int~im}\phi)\cup_{\phi|_{S^p\times S^{q-1}}} (D^{p+1}\times S^{q-1}).$

obtained by surgery, via cutting out the interior of $S^p \times D^q$ and gluing in $D^{p+1}\times S^{q-1}$ along their boundary $\partial(S^p \times D^q)=S^p \times S^{q-1}=\partial(D^{p+1}\times S^{q-1})$. The trace of the surgery

$W := (M\times I) \cup_{S^p\times D^q\times \{1\}} (D^{p+1}\times D^q)$

defines an elementary cobordism $(W;~M,~N)$. Note that M is obtained from N by surgery on $D^{p+1}\times S^{q-1}\subset N$. This is called reversing the surgery.

Every cobordism is a union of elementary cobordisms, by the work of Morse, Thom and Milnor.

Examples

1. The unit line

The unit line [0,1] is a 1-dimensional cobordism between the 0-dimensional manifolds {0}, {1}, and more generally for any closed manifold M there is defined a cobordism $M \times ([0,1];\{0\},\{1\})$ from $M \times \{0\}$ to $M \times \{1\}$.

2. Pair of pants

A cobordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).

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.

More generally, this picture shows that for any two n-dimensional manifolds M,M', the disjoint union $M \sqcup M'$ and connected sum $M \# M'$ are cobordant, as the first corresponds to $S^n \sqcup S^n$ and the latter to $S^n \# S^n \cong S^n.$ The connected sum $M \# M'$ is obtained from the disjoint union $M \sqcup M'$ by surgery on an embedding $S^0 \times D^n \subset M \sqcup M'$, and the cobordism is the trace of the surgery.

3. Surgery on the Circle

Fig. 1

As per the above definition, a surgery on the circle consists of cutting out a copy of $S^0 \times D^1$ and glueing in $D^1 \times S^0$. The pictures in Fig. 1 show that the result of doing this is either (i) S¹ again, or (ii) two copies of S¹.

Fig. 2a

Fig. 2b

4. Surgery on the 2-Sphere

In this case there are more possibilities, since we can start by cutting out either $S^0 \times D^2$ or $S^1 \times D^1$.

• (a) $S^1 \times D^1$: If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in $S^0 \times D^2$ - that is, two disks - and it's clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)

Fig. 2c. This shape cannot be embedded in 3-space.

• (b) $S^0 \times D^2$: Having cut out two disks $S^0 \times D^2$, we glue back in the cylinder $S^1 \times D^1$. Interestingly, there are two possible outcomes, depending on whether our glueing 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 $S^1 \times S^1$, but if they are different, we obtain the Klein Bottle (Fig. 2c).

5. 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 $N:=f^{-1}(c+\epsilon)$ is obtained from $M:=f^{-1}(c-\epsilon)$ by a p-surgery. The inverse image $W:=f^{-1}([c-\epsilon,c+\epsilon])$ defines a cobordism (W;M,N) that can be identified with the trace of this surgery.

Manifolds that are not boundaries

The 2n-dimensional real projective space RP²ⁿ is the simplest example of a (compact) closed manifold that is not the boundary of a manifold: the Euler characteristic of a boundary must be even, but the Euler characteristic of RP²ⁿ is 1. More generally, any product of even-dimensional real projective spaces is not a boundary.

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.

Context

Cobordisms are objects of study in their own right, apart from cobordism classes. Categorically, one can also think of a cobordism (W;M,N) as a map from M to N, and thus define categories of cobordisms. A cobordism is a kind of cospan:^[2] M → W ← N.

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. Equivalently, its cobordism class is trivial.

For example:

• The circle (and more generally, n-sphere) are null-cobordant since they bound an (n+1)-disk.
• Every orientable surface is null-cobordant, because it is the boundary of a handlebody.

Null-cobordisms with additional structure are called fillings. "Bordism" and "cobordism" are sometimes used 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 bord, 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-.

Geometry, and the connection with Morse theory and handlebodies

Given a cobordism (W;M,N) there exists a smooth function $f : W \to [0,1]$ 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 $M \times [0,1]$ by attaching one handle for each critical point of f

The 3-dimensional cobordism $W=S^1 \times D^2 - D^3$ between the 2-sphere M = S² and the 2-torus $N=S^1\times S^1$, with N obtained from M by surgery on $S^0 \times D^2 \subset M$, and W obtained from $M \times I$ by attaching a 1-handle $D^1 \times D^2$.

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.

Development

This section covers cobordism of manifolds with different types of structures, and algebraic topological calculations concerned with these.

Cobordism classes

The general bordism problem is to calculate the cobordism classes of manifolds subject to various conditions.

G-cobordism

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 Stong.^[3]

Unoriented cobordism

For more details on this topic, see List_of_cohomology_theories#Unoriented_cobordism.

The cobordism class $[M] \in \Omega_n^{\text{O}}$ of a closed n-dimensional manifold M is determined by the Stiefel-Whitney characteristic numbers, which depend on the stable isomorphism class of the tangent bundle. Thus if M has a stably trivial tangent bundle then $[M]=0 \in \Omega_n^{\text{O}}$. Every closed manifold M is such that $M \cup M = \partial (M \times I)$, so 2x = 0 for every $x \in \Omega_n^{\text{O}}$. In 1954 René Thom computed

$\Omega_n^{\text{O}}=$Z$_2[x_i \vert i \geq 1, i \neq 2^j -1 ]$

with one generator x_i in each dimension $i \neq 2^j-1$. For even i is possible to choose $x_i=[\mathbb{R}\mathbb{P}^i]$, the cobordism class of the i-dimensional real projective space.

The low-dimensional unoriented cobordism groups are

$\Omega_0^{\text{O}}=\mathbb{Z}_2~,~\Omega_1^{\text{O}}=0~,~\Omega_2^{\text{O}}=\mathbb{Z}_2~,~\Omega_3^{\text{O}}=0~,~\Omega_4^{\text{O}}=\mathbb{Z}_2\oplus \mathbb{Z}_2~,~\Omega_5^{\text{O}}=\mathbb{Z}_2~.$

The mod2 Euler characteristic $\chi(M) \in \mathbb{Z}_2$ of an unoriented 2i-dimensional manifold M is an unoriented cobordism invariant. For example, for any $i_1,i_2,\dots,i_k \geq 1$

$\chi(\mathbb{RP}^{2i_1}\times \mathbb{RP}^{2i_2}\times \dots \times \mathbb{RP}^{2i_k})~=~1~.$

The mod2 Euler characteristic map $\chi:\Omega_{2i}^{\text{O}} \to \mathbb{Z}_2$ is onto for all $i \geq 1$, and an isomorphism for i = 1.

Oriented cobordism

For more details on this topic, see List_of_cohomology_theories#Oriented_cobordism.

Assume all manifolds are oriented. Then an oriented cobordism is a manifold W whose boundary (with the induced orientations) is $M \sqcup (-N)$, where − N denotes N with the reversed orientation.

Why the reversed orientation? Firstly, because as oriented manifolds, the boundary of the cylinder $M \times I$ is $M \sqcup (-M)$: both ends have opposite orientations (this is most familiar for the interval: $\partial I = \{1\}-\{0\}$). It is also the correct definition in the sense of extraordinary cohomology theory. For an oriented closed manifold M it is not in general the case that 2M is an oriented boundary, and the cobordism class $[M] \in \Omega_*^{\text{SO}}$ could have infinite torsion, i.e., such that $k[M] \neq 0$ for every non-zero integer k

The oriented cobordism groups are given modulo torsion by

$\Omega_*^{\text{SO}}\otimes \mathbb{Q}~=~\mathbb{Q}[y_{4i}|i \geq 1]$,

the polynomial algebra generated by the oriented cobordism classes $y_{4i}=[\mathbb{CP}^{2i}] \in \Omega_{4i}^{\text{SO}}$ of the complex projective spaces (Thom, 1952) The oriented cobordism group $\Omega_*^{\text{SO}}$ is determined by the Stiefel-Whitney and Pontrjagin characteristic numbers (Wall, 1960). Two oriented manifolds are oriented cobordism if and only if their Stiefel-Whitney and Pontrjagin numbers are the same.

The low-dimensional oriented cobordism groups are

$\Omega_0^{\text{SO}}=\mathbb{Z}~,~\Omega_1^{\text{SO}}=\Omega_2^{\text{SO}}=\Omega_3^{\text{SO}}=0~,~\Omega_4^{\text{SO}}=\mathbb{Z}~,~\Omega_5^{\text{SO}}=\mathbb{Z}_2~.$

The signature of an oriented 4i-dimensional manifold M

$\sigma(M)~=~\text{signature of the intersection form on}~H^{2i}(M) \in \mathbb{Z}$

is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the Hirzebruch signature theorem.

For example, for any $i_1,i_2,\dots,i_k \geq 1$

$\sigma(\mathbb{CP}^{2i_1}\times \mathbb{CP}^{2i_2}\times \dots \times \mathbb{CP}^{2i_k})~=~1~.$

The signature map $\sigma:\Omega_{4i}^{\text{SO}} \to \mathbb{Z}$ is onto for all $i \geq 1$, and an isomorphism for i = 1.

Cobordism as an extraordinary cohomology theory

Every vector bundle theory (real, complex etc.) has a extraordinary cohomology theory called K-theory. Similarly, every cobordism theory Ω^G has an extraordinary cohomology theory, with homology ("bordism") groups $\Omega^G_n(X)$ and cohomology ("cobordism") groups $\Omega^n_G(X)$ for any space X. The generalized homology groups $\Omega_*^G(X)$ are covariant in X, and the generalized cohomology groups $\Omega^*_G(X)$ are contravariant in X. The cobordism groups defined above are, from this point of view, the homology groups of a point: $\Omega_n^G = \Omega_n^G(pt.)$. Then $\Omega^G_n(X)$ is the group of bordism classes of pairs (M,f) with M a closed n-dimensional manifold M (with G-structure) and $f:M \to X$ a map. Such pairs (M,f), (N,g) are bordant if there exists a G-cobordism (W;M,N) with a map $h:W \to X$, which restricts to f on M, and to g on N.

An n-dimensional manifold M has a fundamental homology class $[M] \in H_n(M)$ (with coefficients in Z₂ in general, and in Z in the oriented case), defining a natural transformation

$\Omega^G_n(X) \to H_n(X);(M,f) \to f_*[M]~,$

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 $\Omega^n_G(X)$ 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

$\Omega^G_n(X)=\sum\limits_{p+q=n}H_p(X;\Omega^G_q(pt.))$.

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). ^[4]

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.

Categories of cobordisms

Cobordisms form a category where the objects are closed manifolds and the morphisms are cobordisms. 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 $(W'\cup_N W;M,P)$.

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 trivial, but the category of cobordism is still interesting. For instance, the disk bounding the circle corresponds to a null-ary operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.

Cobordism of piecewise linear and topological manifolds

Originally, cobordism was introduced for differentiable manifolds. The theory works also for PL and topological manifolds, with bordism groups $\Omega_*^{PL}(X)$, $\Omega_*^{TOP}(X)$, which are generalized homology groups. The cobordism groups of PL and topological manifolds are harder to compute.

See also

• h-cobordism
• Link concordance
• List of cohomology theories
• Symplectic filling

Notes

1. ^ The notation "n + 1-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional cobordism" refers to a 5-dimensional cobordism between 4-dimensional manifolds or a 6-dimensional cobordism between 5-dimensional manifolds.
2. ^ While every cobordism is a cospan, the category of cobordisms is not a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that M and N form a partition of the boundary of W is a global constraint.
3. ^ Stong, Robert E. (1968). Notes on cobordism theory. Princeton University Press. 
4. ^ Ravenel, D.C. (April 1986). Complex cobordism and stable homotopy groups of spheres. Academic Press. ISBN 0125834306 

References

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• M. F. Atiyah, Bordism and cobordism Proc. Camb. Phil. Soc. 57, pp. 200–208 (1961).
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• Kosinski, A. (October 19, 2007). Differential Manifolds. Dover Publications 
• Madsen, Ib; Milgram, R. James (1979). The classifying spaces for surgery and cobordism of manifolds. Princeton, New Jersey: Princeton University Press. ISBN 978 0 69108226 4 
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• L. Pontryagin, Smooth manifolds and their applications in homotopy theory American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959).
• D. Quillen, On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298.
• D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Acad. Press (1986).
• Yu. B. Rudyak (2001), "Cobordism", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/C/c022780.htm 
• Yu. B. Rudyak, On Thom spectra, orientability, and (co)bordism, Springer (2008).
• R. E. Stong, Notes on cobordism theory, Princeton Univ. Press (1968).
• Taimanov, I. A. (2007). Topological library. Part 1: cobordisms and their applications. Series on Knots and Everything. 39. S. Novikov (ed.). World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ. ISBN 978-981-270-559-4; 981-270-559-7. 
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External links

• Bordism on the Manifold Atlas.