Nilpotent subgroups of self equivalences of torsion spaces

LetX be a finitep-torsion based connected nilpotent CW-complex. We give a criterion of a subgroup of ε(X), the group of self equivalences ofX, to be a nilpotent group, in terms of its action onE _*(X), whereE is a CW-spectrum, satisfying some technical conditions.     

The group of self homotopy equivalences of some localized aspherical complexes

By studying the group of self homotopy equivalences of the localization (at a prime p and/or zero) of some aspherical complexes, we show that, contrary to the case when the considered space is a nilpotent, ?^m _#(X_p ) is in general different from ?^m _#(X)p. That is the case even when X = K (G, 1) is a finite complex and/or G satisfies extra finiteness or nilpotency conditions, for instance, when G is finite or virtually nilpotent. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constructing homotopy equivalences

In this paper we study the following construction of homotopy equivalences: Take a codimension one separating submanifold N^n?1 of Mⁿ, cut along N and glue the pieces together by a homeomorphism of N homotopic to the identity. Aside from the question of which homotopy equivalences can be so obtained, we will study qualitative questions such as stability, type of submanifold, etc. Relations to ΣΩ, the oozing problem in surgery theory, and Kervaire classes will be discussed.     

A bound for the nilpotency of a group of self homotopy equivalences

Let be the group of homotopy classes of self-homotopy equivalences of such that . We prove that is a nilpotent group and that .

     

Globalizing weak homotopy equivalences

We generalize the basic quasifibration theorem of Dold and Thom (1958), replacing quasifibrations by weak equivalences.     

Extending local homotopy equivalences

We give an elementary proof of one of tom Dieck’s theorems. The theorem says that iff:X → Y is a local homotopy equivalence in a strong enough sense, thenf is a homotopy equivalence globally. Applications, 1. The base space of any numerable principalG-bundle is of the sme homotopy type as the Borel space of the bundle. 2. The nerve of a numerable coveringU ofX for which all finite intersections are contractible is of the same homotopy type asX.     

R-nilpotency in homotopy equivalences

We study the monoid of self homotopy equivalences of anR-nilpotent space, with the goal of understanding the actions of a cyclic group of orderp on a simply-connected homologically finite space with uniquelyp-divisible homotopy groups. This work was supported in part by the National Science Foundation.     

Obstructions to homotopy equivalences

An obstruction theory is developed to decide when an isomorphism of rational cohomology can be realized by a rational homotopy equivalence (either between rationally nilpotent spaces, or between commutative graded differential algebras). This is used to show that a cohomology isomorphism can be so realized whenever it can be realized over some field extension (a result obtained independently by Sullivan).In particular an algorithmic method is given to decide when a c.g.d.a. has the same homotopy type as its cohomology (the c.g.d.a. is called formal in this case).The chief technique is the construction of a canonically filtered model for a commutative graded differential algebra (over a field of characteristic zero) by perturbing the minimal model for the cohomology algebra. This filtered model is also used to give a simple construction of the Eilenberg-Moore spectral sequence arising from the bar construction. An example is given of a c.g.d.a. whose Eilenberg-Moore sequence collapses, yet which is not formal.     

Globalizing weak homotopy equivalences

We generalize the basic quasifibration theorem of Dold and Thom (1958), replacing quasifibrations by weak equivalences.     

Combinatorial isomorphisms and combinatorial homotopy equivalences

This work properly belongs to combinatorial group theory. But in its motivation and applications, it is concerned with the homotopy theory of two-dimensional cellular spaces. We describe both the combinatorial and homotopical aspects of this work in the following introduction.     

On the group of homotopy equivalences of a manifold

We consider the group of homotopy equivalences of a simply connected manifold which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group for special manifolds , for example if is a connected sum of products of spheres. In particular the group is determined completely. Also the connection of with the group of isotopy classes of diffeomorphisms of is studied.