# 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?

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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