Morphisms of A-infinity Bialgebras and Applications

Details Morphisms of A-infinity Bialgebras and Applications Morphisms of A-infinity Bialgebras and Applications

2011-06-25

arxiv.org/abs/1106.5090v4

Mathematics

Algebraic Topology

37 pages; 6 figures. Version 4 is the (final) version published in J. Homology, Homotopy and Applications. In addition to incl

Scientific paper

We define the notion of a relative matrad and construct a new family of polytopes JJ={JJ(n,m)=JJ(m,n)}, m,n > 0, called bimultiplihedra, of which JJ(n,1)=JJ(1,n) is the multiplihedron J(n). We realize the free relative matrad rH_\infty by extending the free A_\infty-bimodule structure on cellular chains of multiplihedra to a free H_\infty-bimodule structure on cellular chains C_*(JJ). A morphism G : A ==> B of A_\infty-bialgebras is the image of a map C_*(JJ) --> Hom(TA,TB) of relative matrads. We prove that the homology of every A_\infty-bialgebra over a commutative ring with unity (in particular, the bialgebra S_*(\Omega X;Z) of singular chains on Moore base pointed loops) admits an induced A_\infty-bialgebra structure. We extend the classical Bott-Samelson isomorphism to an isomorphism of A_\infty-bialgebras, and identify the A_\infty-bialgebra structure of H_*(\Omega \Sigma X;Q) as the first nontrivial rational homology invariant for \Omega \Sigma X. For each n > 1, we construct a space X_n whose loop cohomology H=H^*(\Omega X_n ;Z_2) comes equipped with a nontrivial induced operation \omega^n_2 : H \otimes H \to H^{\otimes n}. To our knowledge, \omega^n_2 is the first known example of a non-operadic operation on loop cohomology.

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