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The equivariant topological s-cobordism theorem. (English) Zbl 0646.57022

By an h-cobordism (W,M) on an n-manifold M the author means an \((n+1)\)- manifold W, together with an inclusion \(M\subset \partial W\), with bicollared boundary, such that both \(M\subset W\) and \(\overline{\partial W-M}\subset W\) are proper homotopy equivalences, and by an s-cobordism theorem he means a theorem which gives necessary and sufficient conditions for a smooth, PL or topological h-cobordism to be equivalent rel M to the trivial h-cobordism (M\(\times I,M\times 0)\), or more generally a calculation of the set h(M) of isomorphism classes of h- cobordisms on M in the chosen category.
In the paper under review, the author works in the category of topological manifolds with locally linear actions of a comact Lie group G to get calculations of the corresponding sets h(M) of isomorphism classes of G-h-cobordisms on topological locally linear G-manifolds M.
In the introduction of the paper, the author provides a detailed history of related results both in non-equivariant and equivariant categories, including a discussion of G-s-cobordism theorems for smooth and PL G- manifolds. The starting point of the author’s approach form three theorems from the non-equivariant topology due to Kirby and Siebenmann, namely, the Concordance Implies Isotopy Theorem, the Product Structure Theorem, and a direct consequence of the two theorems, the Handlebody Existence Theorem. These results are not true equivariantly and the author’s approach to the topological G-s-cobordism theorem is by identifying the obstructions to the above phenomena in the equivariant case. The obstructions are in the equivariant topological Whitehead group \(Wh_ G^{Top}(M)\) of the G-manifold M in question. The Whitehead group is defined by imposing an equivalence relation among appropriate pairs (Y,M), and the equivalence relation may be defined either in terms of equivariantly cell-like maps or by means of the universal equivariant Hilbert cube \(Q_ G\) which is the product of infinitely, countably many copies of the unit disk in the regular real representation of G.
More specifically, \((Y_ 1,M)\sim (Y_ 2,M)\) if and only if there is a G-homeomorphism \(Y_ 1\times Q_ G\cong Y_ 2\times Q_ G\) which commutes up to proper G-homotopy with the natural inclusions of M. The Whitehead group \(Wh_ G^{Top}(M)\) contains, as a direct summand, the subgroup \(Wh_ G^{Top,\rho}(M)\) generated by pairs (Y,M) such that \(Y\) \(H_{\alpha}-Y_{\alpha}^{>H}=\emptyset\) whenever \(M\) \(H_{\alpha}- M_{\alpha}^{>H}=\emptyset\). Any G-h-cobordism (W,M) represents an element in \(Wh_ G^{Top,\rho}(M)\), called the torsion of (W,M) and denoted by \(\tau\) (W,M). This yields the torsion function \[ \tau: h(M)\to Wh_ G^{Top,\rho}(M)\subset Wh_ G^{Top}(M) \] which turns out to be an isomorphism onto \(Wh_ G^{Top,\rho}(M)\), so that we get the expected calculation of h(M). In particular, a G-h-cobordism (W,M) is topologically trivial if and only if \(\tau (W,M)=0\) in \(Wh_ G^{Top,\rho}(M)\); i.e., there is a G-homeomorphism \(M\times Q_ G\cong W\times Q_ G\) which commutes up to proper G-homotopy with the natural inclusions of M.
The author discusses related results and obtains a number of corollaries including a version of the equivariant annulus conjecture. Moreover, he provides the corresponding results in the isovariant theory. In all results, the G-manifolds fulfill some hypotheses including dimension and/or codimension type conditions. The obtained results depend on the joint work of the author with J. West [Bull. Am. Math. Soc., New Ser. 12, 217-220 (1985; Zbl 0572.57021) and Geometry and Topology, Proc. Conf., Athens/Ga. 1985, Lect. Notes Pure Appl. Math. 105, 277-295 (1987; Zbl 0613.57023)].
Reviewer: K.Pawałowski

MSC:

57S10 Compact groups of homeomorphisms
57N70 Cobordism and concordance in topological manifolds
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References:

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