Non-G-Equivalent Moore G-Spaces of the Same Type

Doman, Ryszard

doi:10.2307/2047132

Proceedings of the American Mathematical Society

1988

DOI: 10.2307/2047132

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Non-G-Equivalent Moore G-Spaces of the Same Type

Ryszard Doman¹

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1989

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“…When R = Q and X H is simply-connected for all H ≤ G, the space X is called a rational Moore G-space. Rational Moore G-spaces are studied extensively in equivariant homotopy theory and many interesting results are obtained on homotopy types of these spaces (see [16] and [11]).…”

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Equivariant Moore spaces and the Dade group

Yalcin

2017

Advances in Mathematics

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Let G be a finite p-group and k be a field of characteristic p. A topological space X is called an n-Moore space if its reduced homology is nonzero only in dimension n. We call a G-CWcomplex X an n-Moore G-space over k if for every subgroup H of G, the fixed point set X H is an n(H)-Moore space with coefficients in k, where n(H) is a function of H. We show that if X is a finite n-Moore G-space, then the reduced homology module of X is an endo-permutation kG-module generated by relative syzygies. A kG-module M is an endo-permutation module if End k (M ) = M ⊗ k M * is a permutation kG-module. We consider the Grothendieck group of finite Moore G-spaces M(G), with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies.

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Equivariant Moore spaces and the Dade group

Yalcin

2017

Advances in Mathematics

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“…In [4] there is given an example of a rational coefficient system M and two Moore G-spaces Li,L 2 of type (M, 2) which are not G-equivalent. In [2] we have shown, by methods completely different from that of [4], that for the system M there exist infinitely many non G-equivalent Moore G-spaces of type (M, 2).…”

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“…The aim of this note is to show that all but one of the Moore G-spaces of type (M,2) constructed in [2] are not co-Hopf G-spaces. Thus, we show that the well known result that every simply connected Moore space is a co-H-space does not hold in the G-equivariant context.…”

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“…in which the action on Q = M(G/e) is trivial. We shall use the following property of the system M. We have observed in [2] that each Moore G-space of type (M,2) has only two non-trivial systems of homotopy groups: 7r 2 (X) = 7r 3 (Z) = M and that this implies that it is determined, up to G-homotopy type, by its equivariant ^-invariant k(X) which lies in the Bredon cohomology group //^(^(M,2),M) [ is the cup-product in Bredon cohomology. We shall use the following facts proved in [2].…”

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“…D REMARK 3.3. We have shown in [2] that, varying u all over the group Ext 2 (£ 2 (/i:(M,2),M), we can obtain infinitely many non G-equivalent Moore Gspaces of type (M, 2). Thus, it follows that there exist infinitely many non G-equivalent Moore G-spaces of type (M, 2) which are not co-Hopf G-spaces.…”

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Moore G-Spaces Which are not Co-Hopf G-Spaces

Doman

1989

Can. math. bull.

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Let G be a finite group. By a Moore G-space we mean a G-space X such that for each subgroup H of G the fixed point space XH is a simply connected Moore space of type (MH,n), where MH is an abelian group depending on H, and n is a fixed integer. By a co-Hopf G-space we mean a G-space with a G-equivariant comultiplication. In this note it is shown that, in contrast to the non-equivariant case, there exist Moore G-spaces which are not co-Hopf G-spaces.

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Non-G-Equivalent Moore G-Spaces of the Same Type

Cited by 2 publications

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Equivariant Moore spaces and the Dade group

Equivariant Moore spaces and the Dade group

Moore G-Spaces Which are not Co-Hopf G-Spaces

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