# Automorphism of first homology and mapping class group

It is known that for a torus $\Sigma$, every automorphism of $H_1(\Sigma; \mathbb{Z})$ is induce by an orientation preserving self-homeomorphism of $\Sigma$ unique up to isotopy. In onther words, there is a bijection between the mapping class group of $\Sigma$ and $Aut(H_1(\Sigma; \mathbb{Z}))$.

Question; Is it still true for a general compact orientable 2-surface $\Sigma$? Or is this special to a torus?

-
This is very unique to the torus. What you are interested in is the mapping class group of the surface, which is quite complicated and an active area of research. I recommend looking at the introduction of the book "A primer on mapping class groups" by Farb and Margalit, available here : math.utah.edu/~margalit/primer – Andy Putman Feb 16 '12 at 17:08
This can be appropriately generalized for closed surfaces. The Dehn-Nielsen-Baer theorem says $MCG(\Sigma)$ is isomorphic to $Out(\pi_1(\Sigma))$. For the torus, we simply get $\pi_1=H_1$ and $Aut=Out$. – Steve D Feb 16 '12 at 20:33

The magic words are "Torelli subgroup" (google, and you will find a million hits) -- that is the kernel of the map from the mapping class group to the automorphism group of the first homology. The torus (I usually think of the punctured torus) is also unique in that for it that map is surjective (the image is, in general, the symplectic group, which is not usually equal to the special linear group except when the dimension is equal to two).

-