1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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 '20 at 23:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.