Recently there has been substantial progress towards computing the mod 2 cohomology of low rank sporadic simple groups. In fact the mod 2 cohomology of every sporadic simple group not containing (Z/2) has been computed, with the notable exceptions of HS (the Higman–Sims group) and Co3 (one of the Conway groups). The reason for these exceptions is that these two groups have large and complicated...

Let G be a compact Lie group. What is the appropriate generalization of singular cohomology to the category of G-spaces XI The simplest choice is the ordinary cohomology of EG xG X, where EG is the total space of a universal principal G-bundle. This Borel cohomology [1] is readily computable and has many applications, but is clearly inadequate for such basic parts of G-homotopy theory as obstru...

A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if hull of every compact subset again compact. In this paper we prove that there exist locally metrizable complete groups which endowed with weak topology associated their character G∧, do not qcp. Thus, Krein’s Theorem, a well known result in framework convex spaces, cannot be fully extended grou...

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group group-like elements is finitely generated. Our strategy has three major steps. first reduce problem to finite generation dimensional Nichols algebras diagonal type. For algebras, we do detailed analysis via Anick resolution reducing further specific combinatorial properties. Finally, chec...

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and A1-homotopy invarianc...