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Consider a space $X$ and the group $Homeo(X)/\sim$ of homeomorphisms on $X$ modulo homotopies which are homeomorphisms in each step. One could also consider diffeomorphisms on $X$ or whatsoever.

Have these groups a special name or is there a name for a theory dealing with these kind of groups? Does anybody have a reference where such groups are computed? Thank you.

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  • $\begingroup$ Following Todd Trimble''s answer below, perhaps you should look into Teichmuller Theory. This certainly covers the mapping class group of some special $X$ and has a very nice (and very popular) theory developed presently. $\endgroup$
    – David White
    Jul 13, 2011 at 21:56

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Perhaps the mapping class group of $X$? There is an extensive theory for mapping class groups and their computations. The mapping class group of the (2-dimensional) torus is $SL_2(\mathbb{Z})$.

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    $\begingroup$ But beware that sometimes MCG is used to mean homotopy equivalences up to homotopy rather than homeomorphisms up to isotopy (or diffeomorphisms up to smooth isotopy). In many important cases (such as most surfaces) the homotopical notion coincides with the other, but in general not. $\endgroup$
    – Tom Goodwillie
    Jul 13, 2011 at 18:03
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  1. How about $\pi_0(\text{Homeo}(X))$?

  2. The papers of Weiss and Williams (automorphisms of manifolds and algebraic $K$-theory...) are relevant since they reduce computations of $\pi_i(\text{aut}(X))$ for $X$ a compact manifold ($\text{aut} = \text{Homeo}, \text{Diff}$) to homotopy theory plus algebraic K-theory in a certain range (the "concordance stable range"). These papers build on results of Hsiang and Anderson.

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