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.

Show description

Read Online or Download Diagram Cohomology and Isovariant Homotopy Theory PDF

Similar group theory books

Representations of Groups: A Computational Approach

The illustration thought of finite teams has noticeable fast progress in recent times with the advance of effective algorithms and computing device algebra platforms. this is often the 1st booklet to supply an advent to the normal and modular illustration thought of finite teams with targeted emphasis at the computational facets of the topic.

Groups of Prime Power Order Volume 2 (De Gruyter Expositions in Mathematics)

This is often the second one of 3 volumes dedicated to easy finite p-group conception. just like the 1st quantity, 1000's of significant effects are analyzed and, in lots of instances, simplified. very important subject matters provided during this monograph comprise: (a) class of p-groups all of whose cyclic subgroups of composite orders are common, (b) category of 2-groups with precisely 3 involutions, (c) proofs of Ward's theorem on quaternion-free teams, (d) 2-groups with small centralizers of an involution, (e) category of 2-groups with precisely 4 cyclic subgroups of order 2n > 2, (f) new proofs of Blackburn's theorem on minimum nonmetacyclic teams, (g) class of p-groups all of whose subgroups of index p² are abelian, (h) class of 2-groups all of whose minimum nonabelian subgroups have order eight, (i) p-groups with cyclic subgroups of index p² are categorized.

Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey

George Mackey was once a rare mathematician of significant strength and imaginative and prescient. His profound contributions to illustration concept, harmonic research, ergodic idea, and mathematical physics left a wealthy legacy for researchers that maintains this day. This publication is predicated on lectures offered at an AMS detailed consultation held in January 2007 in New Orleans devoted to his reminiscence.

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.

Download PDF sample

Rated 4.31 of 5 – based on 29 votes