On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy type as] the classifying space of a finite monoid." The paper was published in 1984. Has any progress been made in the last 30 years that would allow us to replace "it seems likely" by "it is known"?

  • $\begingroup$ Just writing to point out that every path-connected space has the weak homotopy type of $BM$ for some finite monoid $M$. This is a celebrated result of McDuff from 1979. $\endgroup$ – Vidit Nanda Mar 10 '16 at 15:33
  • $\begingroup$ @Vidit Nanda: it seems you wanted to write "for some discrete monoid". $\endgroup$ – BS. Mar 10 '16 at 16:52
  • $\begingroup$ @BS You're absolutely correct, I definitely wanted to write discrete monoid. $\endgroup$ – Vidit Nanda Mar 10 '16 at 17:44
  • $\begingroup$ @ViditNanda Thanks for your comments! Yes, what I wanted was finite (discrete) monoid not just discrete monoid. Simply connectedness is important, since, for example, there's no finite monoid whose classifying space has the weak homotopy type of $S^1$. $\endgroup$ – user46652 Mar 10 '16 at 22:52

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