# Self homotopy equivalence

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?

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