Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a Pontrjagin ring, the localization $$H_*(M)[\pi_0M^{-1}]$$ is isomorphic to $$ H_*(\Omega BM)$$ through the map on homology induced by the canonical map $M\to \Omega BM$. It is a ring isomorphism mapping $\mu_*$ to the product of $H_*(\Omega BM)$.


(1). What is the coproduct structure of the localization $$H_*(M)[\pi_0M^{-1}]$$ induced by $\Delta_*$?

(2). Is the above ring isomorphism $$ H_*(M)[\pi_0M^{-1}]\to H_*(\Omega BM)$$ also an isomorphism of coalgebras preserving the coproduct $\Delta_*$?

  • $\begingroup$ what is the antipode in $H_{\ast}(M)$ ? Is it not just a bialgebra ? $\endgroup$
    – Max
    Aug 22, 2015 at 10:39

1 Answer 1

  1. List item $H_*(M)[\pi _0(M)^{-1}]$ inherits its coalgebra structure from $H_*(M)$. i.e., there is a unique coalgebra structure so that the localization becomes a map of coalgebras. Concretely we "extend" the diagonal by declaring elements of $\pi _0(M)^{-1}$ to be group-like.
  2. List item Since the map $M\rightarrow \Omega BM$ is a map of spaces, it induces a map of coalgebras. So you get an isomorphism of coalgebras.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.