Spines of Teichmuller space of a non-orientable surface

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?

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.

• 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. – Luis Jorge Jul 10 '20 at 23:36