# H-cobordism

A bordism $( W; M _ {0} , M _ {1} )$, where $W$ is a compact manifold whose boundary is the disjoint union of closed manifolds $M _ {0} , M _ {1}$ which are deformation retracts (cf. Deformation retract) of $W$. The simplest example is the trivial $h$- cobordism

$$( M \times [ 0, 1]; M \times \{ 0 \} , M \times \{ 1 \} ).$$

Two manifolds $M _ {0}$ and $M _ {1}$ are said to be $h$- cobordant if there is an $h$- cobordism $W$ joining them.

If $( W; M _ {0} , M _ {1} )$ is an $h$- cobordism such that $W$, $M _ {0}$, $M _ {1}$ are simply-connected differentiable (or piecewise-linear) manifolds and $\mathop{\rm dim} W \geq 6$, then $W$ is diffeomorphic (or piecewise-linearly isomorphic) to $M _ {0} \times [ 0, 1]$: $W \approx M _ {0} \times [ 0, 1]$ and therefore $M _ {0} \approx M _ {1}$( the $h$- cobordism theorem [4]). Thus, proving the isomorphism $M _ {0} \approx M _ {1}$ reduces to providing an $h$- cobordism, which can be achieved by methods of algebraic topology. For this reason, this theorem is basic in passing from the homotopy classification of simply-connected manifolds to their classification up to a diffeomorphism (or a piecewise-linear isomorphism). Thus, if $W ^ {n}$, $n \geq 6$, is a compact differentiable manifold with simply-connected boundary, then it is diffeomorphic to the disc $D ^ {n}$. If $M ^ {n}$, $n \geq 5$, is a manifold that is homotopy equivalent to the sphere $S ^ {n}$, then it is homeomorphic (and even piecewise-linearly isomorphic) to $S ^ {n}$( the generalized Poincaré conjecture).

The $h$- cobordism theorem allows one to classify the differentiable structures on the sphere $S ^ {n}$, $n \geq 5$[6], and also on the homotopy type of an arbitrary closed simply-connected manifold $M ^ {n}$, $n \geq 5$[1].

In the case of an $h$- cobordism $( W; M _ {0} , M _ {1} )$ with $\pi _ {1} W \neq \{ 1 \}$ there is, in general, no diffeomorphism from $W$ to $M _ {0} \times [ 0, 1]$.

All $h$- cobordisms $( W; M _ {0} , M _ {1} )$ with $\mathop{\rm dim} W \geq 6$ and $M _ {0}$ fixed are classified by a certain Abelian group, namely the Whitehead group $\mathop{\rm Wh} \pi _ {1}$ of the group $\pi _ {1} M _ {0}$. Corresponding to a given $h$- cobordism is an element of $\mathop{\rm Wh} \pi _ {1}$ that is an invariant of the pair $( W, M _ {0} )$; it is denoted by $\tau ( W, M _ {0} )$ and is called the torsion (sometimes the Whitehead torsion) of the given $h$- cobordism. If $\tau ( W, M _ {0} ) = 0$( or, equivalently, $\tau ( W, M _ {1} ) = 0$), then the $h$- cobordism is called an $s$- cobordism. If $( W; M _ {0} , M _ {1} )$ is an $h$- cobordism such that $\mathop{\rm dim} W \geq 6$, then $\tau ( W, M _ {0} )$ vanishes if and only if $W \approx M _ {0} \times [ 0, 1]$( the $s$- cobordism theorem). The $h$- cobordism theorem is a special case of this theorem in view of the fact that $\mathop{\rm Wh} \{ 1 \} = 0$. The $s$- cobordism theorem is also true for topological manifolds [9].

For an $h$- cobordism $( W; M _ {0} , M _ {1} )$, the torsion $\tau ( W, M _ {1} )$ is defined along with $\tau ( W, M _ {0} )$; if the given $h$- cobordism is orientable, then $\tau ( W, M _ {0} ) = (- 1) ^ {n - 1 } \tau ^ {*} ( W, M _ {1} )$, where $n = \mathop{\rm dim} W$ and the element $\tau ^ {*}$ is conjugate to $\tau$ in the group $\mathop{\rm Wh} \pi _ {1}$. In particular, if $\pi _ {1}$ is finite and Abelian, $\tau ^ {*} = \tau$.

If two $h$- cobordisms $( W; M _ {0} , M _ {1} )$ and $( W ^ \prime ; M _ {1} , M _ {2} )$ are glued along $M _ {1}$ to the $h$- cobordism $( W \cup W ^ \prime ; M _ {0} , M _ {1} )$, then

$$\tau ( W \cup W ^ \prime , M _ {0} ) = \ \tau ( W, M _ {0} ) + \tau ( W ^ \prime , M _ {1} ).$$

If two copies of $W$ are glued along $M _ {1}$, where $\mathop{\rm dim} W$ is odd and $\pi _ {1} = \mathbf Z _ {5}$, then one obtains an $h$- cobordism $( 2W; M _ {0} , M _ {0} ^ \prime )$, where $M _ {0} = M _ {0} ^ \prime$ when there is no diffeomorphism from $W$ to $M _ {0} \times [ 0, 1]$, that is, when $M _ {0} \approx M _ {1}$ does not imply that the $h$- cobordism connecting them is trivial.

If $M _ {0}$ is a closed connected manifold and $\mathop{\rm dim} M _ {0} \geq 5$, then there exists for any $\tau \in \mathop{\rm Wh} \pi _ {1} M _ {0}$ an $h$- cobordism $( W; M _ {0} , M _ {1} )$ with $\tau ( W, M _ {0} ) = \tau$. If $( W; M _ {0} , M _ {1} )$ and $( W ^ \prime ; M _ {0} , M _ {1} ^ \prime )$( with $\mathop{\rm dim} W \geq 6$) have the same torsion $\tau ( W, M _ {0} ) = \tau ( W ^ \prime , M _ {0} )$, then $W \approx W ^ \prime$ relative to $M _ {0}$. When $\mathop{\rm dim} M _ {0}$ is even and $\pi _ {1} M _ {0}$ is finite, there is a finite set of distinct manifolds that are $h$- cobordant with $M _ {0}$. This is not the case when $\mathop{\rm dim} M _ {0}$ is odd.

If two homotopy-equivalent manifolds $M _ {1}$ and $M _ {2}$ are imbedded in $\mathbf R ^ {N}$, with $N$ sufficiently large, and their normal bundles are trivial, then $M _ {1} \times S ^ {N}$ and $M _ {2} \times S ^ {N}$ are $h$- cobordant. If, moreover, $M _ {1}$ and $M _ {2}$ are of the same simple homotopy type, that is, if the torsion of this homotopy equivalence vanishes, then $M _ {1} \times S ^ {N} \approx M _ {2} \times S ^ {N}$.

If $( W; M _ {0} , M _ {1} )$ is an $h$- cobordism and $P$ is a closed manifold, then there is an $h$- cobordism $( W \times P; M _ {0} \times P, M _ {1} \times P)$ with $\tau ( W \times P, M _ {0} \times P) = \tau ( W, M _ {0} ) \chi ( P)$, where $\chi ( P)$ is the Euler characteristic of $P$. If $\mathop{\rm dim} W \geq 5$ and $P = S ^ {1}$, then

$$W \times S ^ {1} \approx \ M _ {0} \times S ^ {1} \times [ 0, 1] \approx \ M _ {1} \times S ^ {1} \times [ 0, 1].$$

In particular, $M _ {0} \times S ^ {1} \approx M _ {1} \times S ^ {1}$; furthermore, two closed manifolds $M _ {0}$ and $M _ {1}$ of the same dimension $\geq 5$ are $h$- cobordant if and only if $M _ {0} \times \mathbf R ^ {1} \approx M _ {1} \times \mathbf R ^ {1}$.

The $h$- cobordism structure has not been completely elucidated for $n < 6$( 1978). Thus there is the following negative result [8]: There exists an $h$- cobordism $( W; T ^ {4} , T ^ {4} )$, where $T ^ {4}$ is the four-dimensional torus, for which there is no diffeomorphism from $W$ to $T ^ {4} \times [ 0, 1]$; since $\mathop{\rm Wh} \pi _ {1} T ^ {4} = 0$, this means that the $s$- cobordism theorem fails for $n = 5$.

#### References

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