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FIMXII-SCMA2005@AUBURN, Twelfth Annual International Conference on Statistics, Combinatorics, Mathematics and Applications
December 2-4, 2005
Auburn University
Auburn, Alabama, USA

Organizers
Forum for Interdisciplinary Mathematics

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Some general cohomology theories
by
Mahim Ranjan Adhikari
Professor , Department of Pure Mathematics, University of Calcutta, Kolkata-700019, West Bengal,India.
Coauthors: Shibopriya Mitra

Abstract

A functor is called an ordinary cohomology theory if it satisfies all the seven axioms of Eilenberg and Steenrod. These axioms serve to describe the behaviour of the functor on finite CW-complexes. A generalized cohomology theory is a functor which satisfies all the axioms of Eilenberg and Steenrod except for the dimension axiom. We consider a sequence of spaces An together with homotopy equivalences an : An ® WAn+1. Such a sequence A = { An , an } is called an W-spectrum. For example, the Eilenberg-Maclane spectrum { K(G , n) , rn } is that for which An = K(G , n). Spanier proved in (1959) that if Y is a connected CW-complex, then there is a weak homotopy equivalence r : SP¥Y ® WSP¥SY. A theorem of Dold and Thom (1958)implies that the Eilenberg-Maclane space K(G , n) is the infinite symmetric product of a Moore space. In particular, if G = Z (additive group of integer), we have K(Z , n) = SP¥ Sn, which is the infinite symmetric product of the n-sphere Sn. Connecting this with the Hopf's theorem, we get Hn(X ; Z) = [X , K(Z , n)] = [X , SP¥ Sn]. Using these results the authors prove that there is an W-spectrum A for any connected CW-complex Y. The authors also construct the general cohomology theory h*(   ; A) associated with A generalizing the ordinary cohomology theory of Eilenberg and Steenrod. The authors prove that the abelian group of all cohomology operations of degree k for the cohomology theory h*(   ; A) is isomorphic to the group hn+k(SP¥SnY ; A) and the graded abelian group of all stable cohomology operations of degree k for the cohomology theory h*(   ; A) is isomorphic to lim ¬ hn+k(SP¥SnY ; A).

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Date received: May 25, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqt-14.