A simplicial model for the Hopf map
نویسنده
چکیده
We give an explicit simplicial model for the Hopf map S3 → S2. For this purpose, we construct a model of S3 as a principal twisted cartesian product K×ηS, where K is a simplicial model for S1 acting by left multiplication on itself, S2 is given the simplest simplicial model and the twisting map is η : ( S2 ) n → (K)n−1. We construct a Kan complex for the simplicial model K of S1. The simplicial model for the Hopf map is then the projection K ×η S2 → S2. Mathematics Subject Classification : 55U10, 55Q40; 55R10
منابع مشابه
On Posets and Hopf Algebras
We generalize the notion of the rank-generating function of a graded poset. Namely, by enumerating different chains in a poset, we can assign a quasi-symmetric function to the poset. This map is a Hopf algebra homomorphism between the reduced incidence Hopf algebra of posets and the Hopf algebra of quasi-symmetric functions. This work implies that the zeta polynomial of a poset may be viewed in...
متن کاملAn algebraic model for the loop space homology of a homotopy fiber
Let F denote the homotopy fiber of a map f : K → L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L , we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF , the simplicial (Kan) loop group on F . To construct this model, we develop machinery for mod...
متن کاملHomotopy Theory of Simplicial Abelian Hopf Algebras
We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Qu...
متن کاملCombinatorial Hopf Algebras of Simplicial Complexes
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their f -vectors. We al...
متن کاملHomotopy and Homology for Simplicial Abelian Hopf Algebras
Let A be a simplicial bicommutative Hopf algebra over the field F2 with the property that π0A ∼= F2. We show that π∗A is a functor of the André-Quillen homology of A, where A is regarded as an F2 algebra. Then we give a method for calculating that André-Quillen homology independent of knowledge of π∗A. Let G be an abelian group. Since the work of Serre [19] and Cartan [6], we have know that the...
متن کامل