# Download Diagram Cohomology and Isovariant Homotopy Theory by Giora Dula PDF

By Giora Dula

In algebraic topology, obstruction thought offers how to examine homotopy sessions of constant maps by way of cohomology teams; an identical concept exists for sure areas with team activities and maps which are appropriate (that is, equivariant) with recognize to the gang activities. This paintings offers a corresponding atmosphere for yes areas with crew activities and maps which are appropriate in a better experience, referred to as isovariant. the elemental concept is to set up an equivalence among isovariant homotopy and equivariant homotopy for convinced different types of diagrams. outcomes comprise isovariant models of the standard Whitehead theorems for spotting homotopy equivalences, an obstruction thought for deforming equivariant maps to isovariant maps, rational computations for the homotopy teams of convinced areas of isovariant capabilities, and functions to structures and category difficulties for differentiable crew activities.

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**Example text**

4(i). 3, and let Xj (where j > 0) be given as in the proof of the latter result for the closed invariant subset A = 0 (in particular, X$ = 0 ) ; since all isotropy subgroups are normal by hypothesis, each conjugacy class contains exactly one subgroup. We shall prove by induction that / is almost isovariantly homotopic to a map that is isovariant on Xj. The case j = 0 is trivially true by hypothesis, so assume the result for Xj-\. 3 we shall write H = Hj\ since almost isovariant homotopy is a transitive relation on mappings, we may as well assume that / itself is isovariant on Xj-\.

Weinberger [WbJ. A general analysis of isovariant homotopy for continuous locally linear actions will require new techniques because the sets Sing(X / / ) do not necessarily have equivariant mapping cylinder neighborhoods in this case (as before, see [Ql]); it seems quite likely that some sort of sheaf-theoretic machinery will be required. 4 has been more or less understood for some time. For example, in [Har, Prop. 13, p. 24] an equivariant map is equivariantly deformed to an isovariant map by (i) using general position to find an almost isovariant approximation, (ii) deforming the latter to an isovariant map.

3. Let G be a finite group, let X and Y be compact smooth G-manifolds with treelike isotropy structure, let Qx and Qy be canonical quasistratiScations associated to canonical invariant Thorn-Mather stratifications, and suppose that f : X —> Y is an equivariant map that determines a morphism of diagrams B(QFX) —> E(QFY). 3 and obstruction theory in several different ways. 4. Let G be a finite group, let X and Y be compact smooth G-manifolds with treelike isotropy structure, and let B(QFX), B(QFY) and E(QFY) be the usual diagrams associated to canonical quasistratification.