H-cobordism

A bordism , where is a compact manifold whose boundary is the disjoint union of closed manifolds which are deformation retracts (cf. Deformation retract) of . The simplest example is the trivial -cobordism

Two manifolds and are said to be -cobordant if there is an -cobordism joining them.

If is an -cobordism such that , , are simply-connected differentiable (or piecewise-linear) manifolds and , then is diffeomorphic (or piecewise-linearly isomorphic) to : and therefore (the -cobordism theorem [4]). Thus, proving the isomorphism reduces to providing an -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 , , is a compact differentiable manifold with simply-connected boundary, then it is diffeomorphic to the disc . If , , is a manifold that is homotopy equivalent to the sphere , then it is homeomorphic (and even piecewise-linearly isomorphic) to (the generalized Poincaré conjecture).

The -cobordism theorem allows one to classify the differentiable structures on the sphere , [6], and also on the homotopy type of an arbitrary closed simply-connected manifold , [1].

In the case of an -cobordism with there is, in general, no diffeomorphism from to .

All -cobordisms with and fixed are classified by a certain Abelian group, namely the Whitehead group of the group . Corresponding to a given -cobordism is an element of that is an invariant of the pair ; it is denoted by and is called the torsion (sometimes the Whitehead torsion) of the given -cobordism. If (or, equivalently, ), then the -cobordism is called an -cobordism. If is an -cobordism such that , then vanishes if and only if (the -cobordism theorem). The -cobordism theorem is a special case of this theorem in view of the fact that . The -cobordism theorem is also true for topological manifolds [9].

For an -cobordism , the torsion is defined along with ; if the given -cobordism is orientable, then , where and the element is conjugate to in the group . In particular, if is finite and Abelian, .

If two -cobordisms and are glued along to the -cobordism , then

If two copies of are glued along , where is odd and , then one obtains an -cobordism , where when there is no diffeomorphism from to , that is, when does not imply that the -cobordism connecting them is trivial.

If is a closed connected manifold and , then there exists for any an -cobordism with . If and (with ) have the same torsion , then relative to . When is even and is finite, there is a finite set of distinct manifolds that are -cobordant with . This is not the case when is odd.

If two homotopy-equivalent manifolds and are imbedded in , with sufficiently large, and their normal bundles are trivial, then and are -cobordant. If, moreover, and are of the same simple homotopy type, that is, if the torsion of this homotopy equivalence vanishes, then .

If is an -cobordism and is a closed manifold, then there is an -cobordism with , where is the Euler characteristic of . If and , then

In particular, ; furthermore, two closed manifolds and of the same dimension are -cobordant if and only if .

The -cobordism structure has not been completely elucidated for (1978). Thus there is the following negative result [8]: There exists an -cobordism , where is the four-dimensional torus, for which there is no diffeomorphism from to ; since , this means that the -cobordism theorem fails for .

References

[1]	S.P. Novikov, "Homotopy-equivalent smooth manifolds I" Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 365–474 (In Russian)
[2]	J. Milnor, "Lectures on the -cobordism theorem" , Princeton Univ. Press (1965)
[3]	J. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–462
[4]	S. Smale, "On the structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399
[5]	J. Milnor, "Sommes des variétés différentiables et structures différentiables des sphères" Bull. Soc. Math. France , 87 (1959) pp. 439–444
[6]	M. Kervaire, J. Milnor, "Groups of homotopy spheres I" Ann. of Math. (2) , 77 (1963) pp. 504–537
[7]	B. Mazur, "Relative neighbourhoods and the theorems of Smale" Ann. of Math. , 77 (1963) pp. 232–249
[8]	L.C. Siebenmann, "Disruption of low-dimensional handlebody theory by Rohlin's theorem" J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , Topology of manifolds , Markham (1969) pp. 57–76
[9]	R. Kirby, L. Siebenmann, "On the triangulation of manifolds and the Hauptvermutung" Bull. Amer. Math. Soc. , 75 (1969) pp. 742–749
[10]	M.A. Kervaire, "Le théorème de Barden–Mazur–Stallings" M.A. Kervaire (ed.) G. de Rham (ed.) S. Maumary (ed.) , Torsion et type simple d'homotopie , Lect. notes in math. , 48 , Springer (1967) pp. 83–95
[11]	R. Thom, "Les classes caractéristiques de Pontryagin des variétés triangulées" , Symp. Internac. Topol. Algebr. , Univ. Nac. Aut. Mexico & UNESCO (1958) pp. 54–67
[12]	C.P. Rourke, B.J. Sanderson, "Introduction to piecewise-linear topology" , Springer (1972)

Comments

For the generalized Poincaré conjecture see also [a1].

References

[a1]	S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four" Ann. of Math. (2) , 74 (1961) pp. 391–406