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Let $S_{g,n}$ be the orientable surface of genus $g$ and $n$ punctures. Denote $\Gamma_{g,n}$ be the mapping class group of $S_{g,n}$ and $\mathcal T_{g,n}$ the Teichmüller space of $S_{g,n}$.

In http://www.math.lsa.umich.edu/~lji/a-tale-of-two-groups.pdf page 34 item (17) it is mentioned that, provided $n>0$, $\mathcal T_{g,n}$ has a $\Gamma_{g,n}$-equivariant spine of dimension equal to the virtual cohomological dimension of $\Gamma_{g,n}$.

Is there an analogous result for a non-orientable surface with at least one puncture? That is, is it true that the Teichmüller space of a non-orientable surface with at least one puncture has a spine of dimension equal to the virtual cohomological dimension of the corresponding mapping class group?

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Yes... I think so? See Theorem 6.9 of Ivanov's paper Complexes of curves and the Teichmüller modular group (paywalled). It looks like his computation of the vcd goes via a (deformation?) retraction to the desired spine.

By the way, I found this via Ivanov's article in Farb's book Problems on the mapping class group and related topics. See Chapter 4, Section 10.

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  • $\begingroup$ I do not know. I have been reading Ivanov's paper and I am not able to see where does he constructs such a spine. $\endgroup$
    – Luis Jorge
    Jul 10, 2020 at 23:36

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