Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-17T15:14:37.378Z Has data issue: false hasContentIssue false
  • English
  • Français

Co-H-Structures on Moore Spaces of Type (G, 2)

Published online by Cambridge University Press:  20 November 2018

Martin Arkowitz  and
Marek Golasinski
Martin Arkowitz
Affiliation:
Department of Mathematics Dartmouth College, Hanover, New Hampshire 03755, U.S.A.
Marek Golasinski
Affiliation:
Instytut Matematyki UMK, u. Chopina 12/18, 87-100 Torun, Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤm, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤm, 2) and of the homotopy classes of maps from M(ℤm, 2) to M(ℤn, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤm, 2) to M(ℤn, 2) for each co-H-structure on M(ℤm, 2) and on M(ℤn, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤn, 2) on the set of co-H-structures of M(ℤm, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤm, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤm, 2) are associative and non-commutative.

Keywords

55P4555P40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 4 , 01 August 1994 , pp. 673 - 686
Copyright
Copyright © Canadian Mathematical Society 1994

References

[A-L] Arkowitz, M. and Lupton, G., Equivalence Classes of Homotopy-Associative Comultiplications of Finite Complexes, J. Pure Appl. Algebra, to appear.Google Scholar
[Ba] Baues, H.-J., Combinatorial Homotopy and 4-Dimensional Complexes, Expositions in Math. 2, Walter de Gruyter, 1991.Google Scholar
[B-H] Berstein, I. and Hilton, P. J., Suspensions and Comultiplications, Topology 2(1963), 73–82.Google Scholar
[E-G] Eilenberg, S. and Ganea, T., On the Lusternik-Schnirelmann Category of Abstract Groups, Ann. of Math. 65(1957), 517–518.Google Scholar
[Hi] Hilton, P. J., Homotopy Theory and Duality, Notes on Math, and its Applications, Gordon and Breach, 1965.Google Scholar
[H-M-R] Hilton, P. J., Mislin, G. and Roitberg, J., On Co-H-Spaces, Comment. Math. Helv. 53(1978), 1–14.Google Scholar
[H-S] Hilton, P. J. and Stammbach, U., A Course in Homological Algebra, Graduate Texts in Math. 4, Springer-Verlag, 1971.Google Scholar
[Si] Sieradski, A. J., Stabilization of Self-Equivalences of the Pseudoprojective Spaces, Michigan Math. J. 19(1972), 109–119.Google Scholar
[Va] Varadarajan, K., Groups for which Moore Spaces M(TI, 1 ) Exist, Ann. of Math. 84(1966), 368–371.Google Scholar
[Wh] Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Math. 61, Springer-Verlag, 1978.Google Scholar