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A Mod Two Analogue of a Conjecture of Cooke
Author(s) -
Aguadé J.,
Broto C.,
Notbohm D.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610796004449
Subject(s) - mathematics , cohomology , mod , homotopy , conjecture , eilenberg–maclane space , pure mathematics , section (typography) , space (punctuation) , equivariant cohomology , invariant (physics) , algebra over a field , combinatorics , discrete mathematics , homotopy group , computer science , mathematical physics , operating system
The mod two cohomology of the three connective covering of S 3 has the form F 2 [X 2n ] ⊗ E(Sq 1 X 2n ) where x 2 n is in degree 2 n and n = 2. If F denotes the homotopy theoretic fibre of the map S 3 → B 2 S 1 of degree 2, then the mod 2 cohomology of F is also of the same form for n = 1. Notice (cf. Section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove the following.