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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?2012-02-21

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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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    Yes, you are right. I've been spending a lot of time with smooth stuff lately so "diffeo" came out just out of habit.2012-02-21