# Diagram Cohomology and Isovariant Homotopy Theory by Giora Dula

By Giora Dula

In algebraic topology, obstruction concept presents the way to examine homotopy periods of constant maps when it comes to cohomology teams; the same conception exists for definite areas with workforce activities and maps which are suitable (that is, equivariant) with recognize to the gang activities. This paintings presents a corresponding surroundings for definite areas with workforce activities and maps which are appropriate in an improved feel, referred to as isovariant. the fundamental concept is to set up an equivalence among isovariant homotopy and equivariant homotopy for convinced different types of diagrams. results comprise isovariant types of the standard Whitehead theorems for spotting homotopy equivalences, an obstruction concept for deforming equivariant maps to isovariant maps, rational computations for the homotopy teams of convinced areas of isovariant features, and functions to buildings and category difficulties for differentiable staff activities.

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Extra resources for Diagram Cohomology and Isovariant Homotopy Theory

Example text

Let pa and pp be the projections of the componentwise normal bundles. 4(z) there is an isovariant diagram preserving homotopy rel Sing(X) from / to a map f\ such DIAGRAM COHOMOLOGY AND ISOVARIANT H O M O T O PY 33 that fipa — ppfi on S(a). We claim there is also an isovariant homotopy from D(a) to D(/3) rel S(a) U Sing(X) from f\ to a length preserving map. First of all, there is an isovariant homotopy rel S(a) U Sing(X) from / i to a map J2 such that the map \$2 sends ^D(a) into ^D(j3) for every positive integer n; this is true because (i) one can find a sequence of positive real numbers 6n < 1 such that lim<5n = 0 and f\ sends 6nD(a) into ^D(/3) for all n, (ii) there is an ambient isotopy of X rel S(a) U Sing(X) that maps ^D(a) fiber preservingly into SnD(a) for all n.

9. Let G be a finite group, and let X and Y be compact locally linear G-manifolds. Let Qx and QY be regular G-invariant quasistratihcations and let B(QFX) and ^4(QF y ) be defined as before. Then the forgetful map G — isovariant G — isovariant homotopy classes of homotopy classes of continuous isovariant continuous isovariant diagram morphisms maps of spaces B(QFX)->A(QFY) X —>Y is an isomorphism. 9. 10. Let G be a finite group, and let smooth G-manifolds with treelike isotropy structure. 4], and let B(QFX) and before.

Of course, this includes the case where ~ / ( / ) is empty and / is an isovariant map that determines a morphism of diagrams as above. The name "almost isovariant" suggests that isovariant maps should be almost isovariant. However, the relationship is not quite that simple because almost isovariance requires the existence of invariant quasistratifications and specific choices of such structures on the domain and codomain. 2. A). 1, let B\$(QFX) be the diagram of closed subspaces associated 27 28 GIORA DULA AND REINHARD SCHULTZ to the quasistratification Qx(6) f°r <\$ > 0, and let f : X —* Y be a continuous isovariant map.