Certain number on the groups of self homotopy equivalences

2014/08/21 by Ho Won Choi, Choi, Ho Won, Kee Young Lee +1
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT

paper · pdf · doi:10.48550/arxiv.1408.4871

9 pages

arxiv created 2014/08/21 · arxiv updated 2014/08/22

Abstract

For a connected based space X, let [X,X] be the set of all based homotopy classes of base point preserving self map of X and let \E(X) be the group of self-homotopy equivalences of X. We denote by \A_\sharp^k(X) the set of homotopy classes of self-maps of X that induce an automorphism of π_i(X) for i=0,1,⋯,k. That is, [f]∈ \A_\sharp^k(X) if and only if π_i(f):π_i(X)→π_i(X) is an isomorphism for i=0,1,⋯ ,k. Then, \E(X)⊆\A_\sharp^k(X)⊆ [X,X] for a nonnegative integer k. Moreover, for a connected CW-complex X, we have \E(X)=\A_\sharp(X). In this paper, we study the properties of \A_\sharp^k(X) and discuss the conditions under which \E(X)=\A_\sharp^k(X) and the minimum value of such k. Furthermore, we determine the value of k for various spaces, including spheres, products of spaces, and Moore spaces.

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