2010/11/11 by Victor Buchstaber, Jelena Grbic, Buchstaber, Victor +1
Mathematics · #16T05 (Primary) 55P35 #55P40 #55T25 #57T05 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:16T05 #msc:55P35 #msc:55P40 #msc:55T25 #msc:57T05
paper · pdf · doi:10.48550/arxiv.1011.2549
arxiv created 2012/11/23 · arxiv updated 2012/11/26
In this paper we explore new relations between Algebraic Topology and the theory of Hopf Algebras. For an arbitrary topological space X, the loop space homology H_*(ΩΣX; \coefZ) is a Hopf algebra. We introduce a new homotopy invariant of a topological space X taking for its value the isomorphism class (over the integers) of the Hopf algebra H_*(ΩΣX; \coefZ). This invariant is trivial if and only if the Hopf algebra H_*(ΩΣX; \coefZ) is isomorphic to a Lie-Hopf algebra, that is, to a primitively generated Hopf algebra. We show that for a given X these invariants are obstructions to the existence of a homotopy equivalence ΣX≃ Σ2Y for some space Y. Further on, using the notion of Hopf algebras, we establish new structural properties of the cohomology ring, in particular, of the cup product. For example, using the fact that the suspension of a polyhedral product X is a double suspension, we obtain a strong condition on the cohomology ring structure of X. This gives an important application in toric topology. For an algebra to be realised as the cohomology ring of a moment-angle manifold \ZP associated to a simple polytope P, we found an obstruction in the Hopf algebra H_*(ΩΣ\ZP). In addition, we use homotopy decompositions to study particular Hopf algebras.