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Let me write $\mathcal{N}_g$ for the mapping class group of the connect sum of $g$ projective planes. Nathalie Wahl proved that these groups enjoy homological stability, and in O. Randal-Williams …

A group is a category with a single object and all morphisms invertible; an abelian group is a monoidal category with a single object and all morphisms invertible. This can be seen via the trick that …

I think it is sometimes written $\operatorname{GSp}_{2g}(\mathbb{Z})$, and called the group of symplectic similitudes

In the following paper U. Tillmann, The representation of the mapping class group of a surface on its fundamental group in stable homology, Q J Math (2010) 61 (3): 373-380. Ulrike Tillmann studies t …

By coincidence I needed to know something about this recently, and one thing I know is that that Ext group vanishes for $0 < \vert m - n \vert < p - 1$. There is a proof in Lemma 6.1 of my paper "Co …

No, because it is not even true for constant families: let $A$ be an acyclic group, so $H_i(A)=0$ for $i>0$, and $B$ be a group which $A$ acts on interestingly, e.g. $B= F(A)$ is the free group on the …

I don't believe this is true. Let $(G, H) = (\Sigma_3, C_3)$ and $f : C_3 \to \Sigma_3$. Then your square says that $$H^1(C_3;\mathbb{Z}/3) = \mathbb{Z}/3 \longrightarrow H^1(\Sigma_3;\mathbb{Z}/3) = …

No, it is not true. For example, let $\mathbf{Z}/2$ act on $\mathbf{Z}$ by inversion, and $G$ be the semidirect product. Then $\mathbf{Z}$ is a torsion-free finite index normal subgroup of $G$, but on …

In a paper (here) with Soren Galatius, we compute the topological group completions of certain topological monoids made up of moduli spaces of surfaces with various structures. Taking $\pi_0$ of these …

It is zero. This is an application of the "centre kills" trick, which I will state in homology. Trick. Let $M$ be a $G$-module for which there is an element $z$ in the centre of $G$ which acts as $-1 …

The second of your questions is answered in Graham Hope and Ulrike Tillmann "On the Farrell cohomology of the mapping class group of non-orientable surfaces" Proc. Amer. Math. Soc. 137 (2009), no. 1, …

For your second point, the map $$d : \mathrm{Out}(F_n) \longrightarrow GL_n(\mathbb{Z}) \overset{\mathrm{det}}\longrightarrow \mathbb{Z}^\times$$ (given by the action on $\mathbb{Z}^n = H_1(F_n;\math …

Let $P_\bullet \to \mathbb{Z}$ be a projective resolution as a $\mathbb{Z}[G]$-module, $\Delta : P_\bullet \to P_\bullet \otimes P_\bullet$ an approximation of the diagonal, and $\phi : P_\bullet \to …