Celebratio Mathematica — Thomas

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Works connected to William Browder

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W. Browder and E. Thomas: “Axioms for the generalized Pontryagin cohomology operations,” Q. J. Math., Oxf. II. Ser. 13 : 1 (1962), pp. 55–60. MR 0140103 Zbl 0104.39701 article

W. Browder and E. Thomas: “On the projective plane of an \( H \)-space,” Ill. J. Math. 7 : 3 (1963), pp. 492–502. MR 0151974 Zbl 0136.43905 article

Let \( G \) be a topological group. Milnor [1956] defines a sequence of principal \( G \)-bundles \( (E_n,B_n,G) \) (\( 1 \leq n < \infty \)) such that \[ SG = B_1 \subset B_2 \subset \cdots \subset B_{\infty} = B_G, \] where \( SG \) denotes the suspension of \( G \) and \( B_G \) is a classifying space for \( G \). The work of Borel [1953; 1954] gives relations between the cohomology of \( G \) and that of \( B_G \), whereas Rothenberg [1961] investigates the cohomology of the spaces \( B_n \).

Suppose now that \( X \) is an \( H \)-space, that is, \( X \) has a continuous multiplication with unit. One then may not be able to define a classifying space \( B_X \), but Stasheff [1961] has defined the projective plane of \( X \), \( P_2X \), which has the homotopy type of the space \( B_2 \) in case \( X \) is actually a group.The purpose of this paper is to discuss the relationship between the cohomology of \( X \) and that of \( P_2X \).