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Please help me find a mistake and resolve a paradox:

Let $S$ be an orientable surface of genus $g\ge 2$ with $n\ge 1$ boundary components. Consider $T_\partial$, the Dehn twist about a curve parallel to some boundary component $\partial$. There are two facts:

1) The mapping class group of $S$ is generated by the Dehn twists about nonseparating simple closed curves. (See Corollary 4.16 in Farb & Margalit's book: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf )

2) A mapping class $f$ commutes with the Dehn twist $T_c$ about a curve $c$ if and only if $f(c)$ is isotopic to $c$ or $\bar c$. (See Fact 3.8 in http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf )

An immediate corollary:

$T_\partial$ is central in the mapping class group of $S$.

Indeed, for any nonseparating simple closed curve $c$, $c$ and $\partial$ are disjoint, hence $T_c(\partial)\simeq\partial$, and so $T_\partial$ commutes with all generators of the mapping class group.

However, this corollary is in contradiction with the following facts:

3) The center of the mapping class group of $S$ is trivial ( p. 77 in http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf ) [UPDATE: This is true for surfaces with punctures, but not with boundary components!]

4) $T_\partial$ is a nontrivial mapping class (Proposition 3.1 in http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf).

I am very confused now.

mathreader
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1 Answers1

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I guess I found where I was confusing myself. The fact 3), as cited from Farb and Margalit's book, deals with surfaces with punctures, but not with surfaces with boundary components. For the latter, it is indeed true that boundary twists are central, and hence the center is isomorphic to $\mathbb Z^n$, if $n$ is the number of boundary components. Let me quote the sentence on p. 77 of Farb and Margalit's book:

"By the same argument, for a surface with boundary, the Dehn twist about any boundary component is central."

mathreader
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