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Given $X$, a simply connected CW-complex of finite type, ${\rm aut}(X)$ denotes the set of its self homotopy equivalences, that are maps $f: X\rightarrow X$ which admits a homotopy inverse (i.e., ${\rm aut}(X)$ is the set of automorphism of $X$ in the pointed homotopy category). ${\rm Baut}(X)$ denotes the associated classification space.

Question: Did we know any thing about the cases ${\rm Baut}(X)$ formal or ${\rm Baut}(X)$ coformal?

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  • $\begingroup$ $\pi_0 aut(X)$ is the automorphism group of $X$ in the homotopy category. $\endgroup$
    – Fernando Muro
    Commented Nov 26, 2014 at 14:16
  • $\begingroup$ I want to know if the formality of ${\rm Baut}(X)$ was already studied elsewhere $\endgroup$
    – MyIsmail
    Commented Nov 26, 2014 at 14:28
  • $\begingroup$ See people.sju.edu/~smith/pdf_files/factor3.pdf, "Rational Type of Classifying Spaces for Fibrations" by Samuel Bruce Smith. $\endgroup$
    – Igor Belegradek
    Commented Nov 26, 2014 at 14:30
  • $\begingroup$ Have you checked J.W.Rutter: Spaces of homotopy self-equivalences - a survey. Springer LNMA 1662 (1997)? $\endgroup$
    – Matthias Wendt
    Commented Nov 26, 2014 at 15:02
  • $\begingroup$ @MatthiasWendt: Thanks, I will do it right now $\endgroup$
    – MyIsmail
    Commented Nov 26, 2014 at 21:04

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