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When I read a text book, I encountered the sentence

"The modular group of genus $n$ is the group of isotopy classes of degree $1$ self-homeomorphism of a closed oriented surface of genus $n$".

Is "degree $1$" equivalent to "orientation preserving homeomorphism"?

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Which textbook is this from? What page is it on? – Jesse Madnick Feb 21 '12 at 5:09
up vote 1 down vote accepted

Well, no. Maps of degree 1 need not be self-homeomorphisms, since they can easily fail to be injective. Imagine taking a circle and looping a little piece of it over itself. This is isotopic to the identity, but is certainly not a self-homeomorphism.

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Then can we say "degree 1"="orientation preserving"? – Link S Feb 21 '12 at 5:57
Recall that both orientation and degree can be defined using the top homology (with integral coefficients, say) of the $n$-manifold $M$ we are considering. An orientation of $M$ is a choice of generator of $H_n(M) = \mathbb Z$, while the degree of a map $f: M \rightarrow M$ is the induced action on top homology. EDIT: I've screwed some things up. Hold on... – NKS Feb 21 '12 at 6:04
Okay, it won't let me keep editing it. Continuing: ...A diffeomorphism $M \to M$ is orientation-preserving if it sends our choice of generator to itself. So any orientation-preserving diffeo is degree 1, but it only makes sense to talk about preserving orientation when we're talking about a diffeo. – NKS Feb 21 '12 at 6:11
@NKS I don't see why you need to take diffeo. The same definition works fine for a homeomorphism. A homeomorphism has degree $1$ or $-1$, and it is orientation preserving iff its degree is $1$. – user17786 Feb 21 '12 at 13:55
Yes, you are right. I've been spending a lot of time with smooth stuff lately so "diffeo" came out just out of habit. – NKS Feb 21 '12 at 16:06

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