Homotopy Classes Of Self-homotopy Equivalences Research Articles

On the realizability of group actions

Let the pair (G,M)R denote a group G acting on an R-module M, R a unitary ring. We ask for the existence of an R-local space X such that (E(X),πk(X))R is equivalent, in a natural way, to (G,M)R, for some k≥2, where E(X) denotes the group of homotopy classes of self-homotopy equivalences of X. If such an X exists, we say that X realizes the group action (G,M)R. We prove that if G is finite and acts faithfully on a finitely generated Q-module M, there exist infinitely many rational spaces realizing (G,M)Q. Our proof relies on providing a positive answer to Kahn's realizability problem for a large class of orthogonal groups that strictly contains finite ones. As a matter of fact, we enlarge the class of groups that is known to be realizable in the classical Kahn's sense.

The group of self-homotopy equivalences of the m-fold smash product of a space

Let E(X) be the set of homotopy classes of self-homotopy equivalences of a space X. The set E(X) is a group by composition of homotopy classes. We study the group E(X∧m) for the m-fold smash product X∧m. We show that the two obvious homomorphisms φ:Sm→E(X∧m) and ψ:E(X)m→E(X∧m) define a homomorphism Ψ:E(X)m⋊Sm→E(X∧m) for any space X, where E(X)m⋊Sm is the semi-direct product of the product group E(X)m by the symmetric group Sm. We show that in most cases the homomorphism φ:Sm→E(X∧m) is a monomorphism and the kernel of Ψ is isomorphic to the kernel of ψ. The injectivity of Ψ is established for the complex projective n-space CPn(n≥2), and hence, E((CPn)∧m) contains a subgroup isomorphic to {±1}m⋊Sm. Sufficient conditions for Ψ to be injective are obtained for the Eilenberg–MacLane complex K(Ar,n) where A is a subring of Q or a ring Z/k(k≥2) and Ar is the free A-module of rank r. From this result, we see that E(K(Ar,n)∧m) contains a subgroup isomorphic to GLr(A)m⋊Sm in many cases.

THE GROUP OF SELF-HOMOTOPY EQUIVALENCES OF A SIMPLY CONNECTED AND 4-DIMENSIONAL CW-COMPLEX

Let X be a CW complex, E(X) the group of homotopy classes of self-homotopy equivalences of X and E∗(X) its subgroup of the elements that induce the identity on homology. This paper deals with the problem 19 in [Contemp. Math., 519 (2010), 217-230]. Given a group G, find a space X such that E(X) E∗(X) = G. For a simply connected and 4-dimensional CW-complex X we define a group B 4 ⊂ aut(H∗(X,Z)) in term of the Whitehead exact sequence of X and we show that this problem has a solution if G ∼= B 4 for some space X.

GOTTLIEB GROUPS AND SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES

Let <TEX>$\varepsilon_#(X)$</TEX> be the subgroups of <TEX>$\varepsilon(X)$</TEX> consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and <TEX>$\varepsilon_*(X) $</TEX> be the subgroup of <TEX>$\varepsilon(X)$</TEX> that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup <TEX>$\varepsilon_#(X)\cap\varepsilon_*(X)\;of\;\varepsilon(X)$</TEX>. We also give some relations between <TEX>$\pi_n(W)$</TEX>, as well as a generalized Gottlieb group <TEX>$G_n^f(W,X)$</TEX>, and a subset <TEX>$M_{#N}^f(X,W)$</TEX> of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of <TEX>$\varepsilon(X)$</TEX> consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.

GROUPS OF SELF-EQUIVALENCES OF SUSPENDED REAL PROJECTIVE SPACES

The group consisting of the based homotopy classes of self-homotopy equivalences is called the self-equivalence group. We determine the group structures of self-equivalence groups, for the suspended real projective space whose dimension is less than or equal to six. The method is to study the multiplicative structure of self-homotopy set induced from the composition of maps. Finding out the invertible element of this monoid give almost all structures of self-equivalence groups. The group of the 1-fold suspension of the four-dimensional real projective space which is not determined similarly is obtained by the another method thought of from Rutter's paper.

Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups

For a based, 1-connected, finite CW-complex X, we study the following subgroups of the group of homotopy classes of self-homotopy equivalences of X: ε ∗(X) , the subgroup of homotopy classes which induce the identity on homology groups, ε ∗(X) , the subgroup of homotopy classes which induce the identity on cohomology groups and ε # dim + r ( X), the subgroup of homotopy classes which induce the identity on homotopy groups in dimensions ⩽ dim X + r. We investigate these groups when X is a Moore space and when X is a co-Moore space. We give the structure of the groups in these cases and provide examples of spaces for which the groups differ. We also consider conditions on X such that ε ∗(X) = ε ∗(X) and obtain a class of spaces (including compact, oriented manifolds and H-spaces) for which this holds. Finally, we examine ε # dim + r ( X) for certain spaces X and completely determine the group when X = S m × S n and X = CP n ∨ S 2 n .

A Note on the Nilpotency of Subgroups of Self-Homotopy Equivalences

Let X be a space that has the homotopy type of a finite simply connected CW complex. We denote by E(X) the group of homotopy classes of self-homotopy equivalences of X. This group has been extensively studied (see [1] for a survey). In this paper we consider the subgroup En#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on the homotopy groups πi(X) for i⩽n, and the subgroup E#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on all the homotopy groups. Our first result is as follows. 1991 Mathematics Subject Classification 55P62, 55P10.

Automorphisms of Spaces with Finite Fundamental Group

Let $X$ be a finite CW-complex with finite fundamental group. We show that the group ${\text {aut}}(X)$ of homotopy classes of self-homotopy equivalences of $X$ is commensurable to an arithmetic group. If in addition $X$ is an oriented manifold then the subgroup ${\text {au}}{{\text {t}}_t}(X)$ of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group. Moreover if $X$ is a smooth manifold of dimension $\geqslant 5$ then the subgroup ${\text {diff}}(X)$ of ${\text {aut}}(X)$ the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group.

Isotropy groups of homotopy classes of maps

Let aut ⁡ ( X ) \operatorname {aut}(X) be the group of homotopy classes of self-homotopy equivalences of a space X X and let [ f ] ∈ [ X , Y ] [f] \in [X,Y] be a homotopy class of maps from X X to Y Y . The aim of this paper is to prove that under certain nilpotency and finiteness conditions the isotropy group aut ⁡ ( X ) [ f ] \operatorname {aut}{(X)_{[f]}} of [ f ] [f] under the action of aut ⁡ ( X ) \operatorname {aut}(X) on [ X , Y ] [X,Y] is commensurable to an arithmetic group. Therefore aut ⁡ ( X ) [ f ] \operatorname {aut}{(X)_{[f]}} is a finitely presented group by a result of Borel and Harish-Chandra.

Self-homotopy equivalences of group cohomology spaces

The group of homotopy classes of self-homotopy equivalences of a space with only two nonvanishing homotopy groups is computed by means of a differential in a Lyndon spectral sequence.

Topological, affine and isometric actions on flat riemannian manifolds II

Explicit examples of finite subgroups of the group of homotopy classes of self-homotopy equivalences of some flat Riemannian manifolds which cannot be lifted to effective actions are given. It is also shown that no finite subgroups of the kernel of π 0(Homeo( M))→Out π 1( M) can be lifted back to Homeo( M), for a large class of flat manifolds M. Some results of an earlier paper by the authors are refined and related to recent work of R. Schoen and S.T. Yau.

The group of self-homotopy equivalences of principal three sphere bundles over the seven sphere

1. The set ℰ (X) of homotopy classes of self-homotopy equivalences of a space X forms a group under composition. In this note I complete the calculation of ℰ (X) when X is a torsion free rank 2 h-space, and give ℰ (X) up to extension for the remaining principal S3 bundles over S7.