# Has this kind of question in topology a special name?

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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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. –  David White Jul 13 '11 at 21:56

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 $SL_2(\mathbb{Z})$.
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.