The group of homotopy equivalences

Kahn, Donald W.

doi:10.1007/bf01112204

Cited by 16 publications

(6 citation statements)

References 3 publications

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“…he a Postnikov system for X. Recall (see [6]) that there is an exact sequence Here f. is the n 'h''induced map" and c~ acts as a coefficient homomorphism.…”

Section: Theorem Let X Be a 1-connected Space With A Finite Number Omentioning

confidence: 99%

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confidence: 99%

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Realization problems for the group of homotopy classes of self-equivalences

Kahn

1976

Math. Ann.

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The group of homotopy classes of homotopy equivalences from a space X to itself, which we write as G(X), has been studies in recent years by Arkowitz and Curjel [1,2], the author [6,7] and several others. It is a natural quotient of the associative H-space made from homotopy-equivalcnces from X to itself under composition, and it has the same position, in the homotopy category, as the group of automorphisms of a group in the category of groups. It also has the practical use of determining how much the choice of k-invariants overdetermines the homotopytype of a space. G(X) is usually a complicated, non-abelian group, but we often have strong information -for example, we know in important cases, that it is finitely-presented (see [7]).Many simple questions are unfortunately not answered. For example: When does G(X) have torsion? When is G(X) finite? One basic problem, due to M.Arkowitz, asked whether there is a reasonable space X with G(X)=Z, the group of integers. The only case where we have the full picture is the fact that G(K(~, n)) = Aut (g).Our goals here are the following:1) We give a characterization of when G(X) is finite, when X is a reasonable space with finite k-invariants. Our results cover H-spaces and generalize results in [1] and [2].2) We construct a space, X, with non-trivial rational homology, for which G(X) is trivial. This shows that there is no reasonable analogue of ~-theory for G(X), even when ~ is a collection of finite groups. It also shows that one cannot expect to deduce the result of 1) above from the corresponding theorem for a product of a finite number of K(GI, 2ni + 1) spaces. We also indicate how one may get a finite complex with the same properties as X.3) We construct a reasonable space X, which has four non-vanishing homotopy groups, for which G(X)= Z.4) We prove two easy propositions which show that the situation in 2) or 3) above cannot happen in the usual stable cases.5) Finally, we indicate various open questions in this area, which should be within reach of present methods.

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“…he a Postnikov system for X. Recall (see [6]) that there is an exact sequence Here f. is the n 'h''induced map" and c~ acts as a coefficient homomorphism.…”

Section: Theorem Let X Be a 1-connected Space With A Finite Number Omentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

Realization problems for the group of homotopy classes of self-equivalences

Kahn

1976

Math. Ann.

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“…Hence the higher homotopy of T is reflected in Ω(T) which is also the fibre of ί*: <F, F\-+ (F', F) o . The study of π λ (T) requires similar consideration [5], [8], [9]. We will consider these groups again in §3 for some special cases.…”

Section: Moreover I*d F and D Induce Isomorphisms In Homotopymentioning

confidence: 99%

On the structure ofB_∞(F), Fa stable space

Siegel¹

1978

Pacific J. Math.

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“…The discussion was an application of their results on homotopy classification. The first papers which dealt exclusively with the group of self-equivalences appeared in 1964 and were due to Kahn [11], Shih [18], and Arkowitz and Curjel [2,3]. The first general results relating the group ¿?…”

Section: Introductionmentioning

confidence: 99%