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I am in the process of learning about Mapping class groups. At this point it seems like most of what I've read involves very low dimensional (surfaces and 3-manifolds) applications.

I was wondering if they were studied (or arise naturally) in higher dimensional settings?

In particular, any references to their uses in homotopy theory would be appreciated.

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2 Answers 2

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In high dimensions there are several variants that are all distinct (which for surfaces they all agree). There's mapping class groups in the "homotopy category" meaning the homotopy-classes of homotopy equivalences of a topological space, with composition giving the group structure. This is a "core" object of study of classical algebraic topology. In the topological/pl/smooth categories there are isotopy classes of homeomorphisms/pl automorphisms/diffeomorphisms of a manifold. The smooth category gets quite a bit of attention -- for example the smooth category mapping class group of $S^n$ (if you restrict to orientation-preserving diffeomorphisms) is the group of homotopy $(n+1)$-spheres, provided $n \geq 5$. There has been some work on stable high-dimensional mapping class groups by people like Giansiracusa (Swansea).

I have to head out but I can add more later.

The Giansiracusa reference is this: http://www.arxiv.org/abs/math.gt/0510599

Modulo some qualifiers the statement is that the stable mapping class group of a 4-manifold is the automorphisms of homology that preserve the intersection form.

Mapping class groups of a products of circles $(S^1)^n$ in the topological, PL and smooth categories were computed by Hatcher in his "Higher simple homotopy theory" paper.

Is there anything in particular you're interested in?

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Thank you for the reference to Giansiracusa. I am looking at some of his papers now. Did you have any certain paper(s) in mind when you referenced him? If you have time to elaborate on your last statement that would be fantastic! –  confusedmath Feb 21 '11 at 19:56
    
Thank you for the references, You've more than answered my question. –  confusedmath Feb 23 '11 at 5:09

As you mentioned, people have been studying the mapping class groups mostly in connection with (two-dimensional) surfaces, and also some work have been done for 3-manifolds - here the most notable result is Kojima's theorem which states that every finite group is the mapping class group of a compact hyperbolic 3-manifold.

I know only a very few cases, where mapping class groups appear in higher dimensions:

One is a recent generalization of Kojima's theorem for higher dimensional hyperbolic manifolds by Belolipetsky and Lubotsky. See: M. Belolipetsky and A. Lubotzky, Finite Groups and Hyperbolic Manifolds, Invent. Math. 162 (2005) 459–472.

Another one is the use of mapping class groups (of 2d surfaces) in the study of symplectic 4-manifolds (and even 6-manifolds) in http://math.berkeley.edu/~auroux/papers/mcg-farb.pdf

All the best, Zoltan

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interesting paper! thanks –  confusedmath Feb 23 '11 at 5:11

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