# elements of $SL(2,\mathbb{Z})$ and their negative multiples acting with the mapping class group of the two-torus $T^2$

Given some element A of $SL(2,\mathbb{Z})$ which is associated with a particular orientation preserving homeomorphism up to isotopy (by the isomorphism $\phi$), what orientation preserving homeomorphisms up to isotopy will the negative multiple of A in $SL(2,\mathbb{Z})$, that is $-1* A$ correspond to with respect to A? For instance, when considering the generators of the fundamental group of the torus, do both A and $-1*A$ "act" on these two generators similarly? Thank you.

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PSL(2, Z) doesn't sit inside SL(2, Z) as a subgroup, at least not in a natural way. It's a quotient. –  Qiaochu Yuan Feb 12 '11 at 15:04
Ah, that was silly, I'll make an edit. –  user6977 Feb 12 '11 at 22:51