Cobordism

A cobordism from a closed oriented $n$-manifold $X_0$ to $X_1$ is a compact oriented $(n+1)$-manifold $W$ whose oriented boundary is $-X_0\bigsqcup X_1$.

Two closed oriented rational Witt spaces are Witt bordant if their signed disjoint union is the collared boundary of a compact oriented Witt space with collared boundary. Disjoint union gives the bordism group ${\Omega }_n^{\mathrm{Witt}}$.

Every odd-dimensional closed oriented Witt space bounds its closed cone: the new vertex has odd-dimensional link, so introduces no Witt obstruction. For a closed oriented two-dimensional pseudomanifold, normalize to a surface, fill each surface component by a handlebody, and attach a collar of the original space using the finite normalization map. Along the singular collar arcs the links are unions of circles; at their interior starting points the links are wedges of two-spheres. These links have zero middle intersection homology, so the construction is a three-dimensional Witt null-bordism.

For $k\ge 1$, the rational middle intersection pairing induces an isomorphism ${\Omega }_{4k}^{\mathrm{Witt}}\cong W(\mathbb{Q})$. The restriction $k>0$ matters: the degree-zero group is $\mathbb{Z}$. In positive dimension congruent to zero modulo four, a space is Witt null-bordant precisely when its middle intersection form is metabolic.

A nontrivial product of positive-dimensional closed oriented Witt spaces of total dimension four has metabolic rational middle intersection form. For dimensions $1+3$, the two Künneth summands in degree two pair only with one another, so either is a half-dimensional isotropic subspace. For dimensions $2+2$, the outside degree-two summands pair with one another, and the $I{H}^1\otimes I{H}^1$ summand has the tensor pairing of two symplectic forms. A Lagrangian in the first symplectic factor tensored with the entire second factor is a half-dimensional isotropic subspace of that summand. Adding one outside summand gives a Lagrangian in the complete middle form. The Witt bordism classification then makes the product null-bordant.

A cobordism is trivial when it is diffeomorphic to a cylinder with the prescribed incoming boundary identification. Both boundary inclusions are homotopy equivalences and its relative homology with respect to either end vanishes. It can have critical points for a chosen cobordism Morse function; those must form an acyclic relative Morse chain complex.

A compact smooth cobordism is an h-cobordism when both boundary inclusions $M_i\hookrightarrow W$ are homotopy equivalences. The name does not itself impose simple connectivity. A trivial cobordism is an h-cobordism because a product cylinder retracts onto either end.

A compact smooth h-cobordism of dimension at least six between connected simply connected closed manifolds is diffeomorphic to a product, relative to its incoming boundary. The boundary manifolds consequently are diffeomorphic. The dimension and simple-connectivity hypotheses are sufficient conditions, not necessary properties of a product cylinder. The two boundary inclusions being homotopy equivalences are necessary for a trivial cobordism.

Framed cobordism is the generalized cohomology theory of stable cohomotopy. Its geometric maps have a stable framing of a vector bundle on their normal data. A finite covering has canonically framed zero-dimensional relative tangent data; its Gysin class of $1$ is the stable transfer of the cover. A trivial $d$-sheeted cover gives $d\cdot 1$ in degree zero.

The degree-zero framed cobordism class is defined by the transfer of the subset cover of a framed vector bundle. With the canonical relative framing it equals $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\cdot 1$ for a globally ordered framing of rank $n$. Put $l_0=1$ and $l_k=0$ outside the rank range. Disjoint-union additivity and fiber-product multiplicativity of transfer give $l_k(\eta \oplus \zeta )={\sum }_{i+j=k}{l}_i(\eta )l_j(\zeta )$.

Complex cobordism is the generalized cohomology theory represented by the stable Thom spaces of universal complex vector bundles. Its geometric cycles use maps with chosen stable complex normal data. Its complex Thom classes give Chern classes in complex cobordism, and proper oriented maps have a Gysin map in complex cobordism.

The displayed product applies when $k+m\ge n$. Move one projective subspace by a unitary isotopy into transverse position. The intersection is a linear projective subspace of dimension $k+m-n$, with normal complex bundle the sum of the restricted input normal bundles. If $k+m<n$, representatives can be made disjoint and the product is zero.

Pulling the external product of two normally complex submanifolds back along the diagonal gives their transverse intersection. Its normal complex structure is the direct sum of the restricted normal complex structures. Multiplication of Thom classes for a direct sum identifies this geometric representative with the cup product in complex cobordism. Empty intersection represents zero.

// Target: geometry-and-topology.bigb

For a rank-$r$ real bundle with chosen stable complex normal orientation, the complex-cobordism Thom class gives the displayed isomorphism by $a\mapsto {\pi }^{\ast }a{u}_{\xi }$. Stabilizing by a trivial complex bundle suspends the Thom space and its class compatibly. The class of an actual complex rank-$n$ bundle has degree $2n$.

Represent a map $f:X\to Y$ by a smooth map, embed its graph into $Y\times {\mathbb{R}}^N$, and specify a stable complex structure on the normal bundle of that embedding, with the standard framing on the auxiliary real coordinates. Stabilization and homotopy identify equivalent choices. The auxiliary dimension is chosen to make the normal rank even; odd relative dimensions use an extra real suspension coordinate. The resulting oriented stable normal data defines the Gysin map in complex cobordism.

For a proper map with fixed complex normal orientation, apply the Thom isomorphism in complex cobordism, pull back by the collapse of a tubular neighborhood of a graph embedding, and desuspend the auxiliary Euclidean coordinates. The degree increase is the relative real codimension. Changing the orientation can change this map.

Restricting the tubular-neighborhood collapse of an embedding back to its source is the normal bundle's zero section. The Thom isomorphism in complex cobordism therefore makes pullback after the Gysin map multiplication by the Euler class in a generalized cohomology theory. For an actual complex normal bundle this is its top Chern class in complex cobordism.

The Milnor-Novikov coefficient theorem identifies the coefficient ring of complex cobordism as the displayed graded polynomial ring. In cohomological grading it has no positive-degree or odd-degree coefficients. Homological grading reverses the signs: $M{U}_{\ast }$ has polynomial generators in degrees $2i$.

An oriented knot cobordism is a compact oriented smooth surface properly embedded in $S^3\times [0,1]$ with prescribed knots at its incoming and outgoing boundary. A connected genus-zero knot cobordism between knots is a knot concordance. Composing knot cobordisms adds their genera.

Start with $(X_0\bigsqcup X_1)\times [0,1]$ and attach an $(n+1)$-dimensional one-handle $D^1\times D^n$ to one $n$-ball in each component of its outgoing boundary. The new outgoing boundary is $X_0\#X_1$, so this is a cobordism from $X_0\bigsqcup X_1$ to their connected sum.