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Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same aspherical 3-manifold.]

The mapping class group $\Gamma = \pi_0\left(\operatorname{Diff}(M;\text{fixing basepoint})\right)$ acts on $\pi_2(M)$. I am wondering about the group of coinvariants $\pi_2(M) // \Gamma := \pi_2(M) / \left(\text{Subgroup generated by the set } R\right)$ where $$R= \left\{(a - \gamma \cdot a) \in \pi_2(M) \text{ for } \gamma \in \Gamma,\, a\in \pi_2(M) \right\}.$$ Is it known what this group is? More specifically, can one tell whether the subgroup generated by the element $[S_1]$ (defined below) is $\mathbb{Z}/g$ or smaller?

What I know

  1. $\pi_2(M)$ itself is a free abelian group.

    McCullough proves in his "Connected sums of Aspherical Manifolds" paper that the universal cover of $M$ is homotopy equivalent to $\bigvee_{\rho \in \pi_1(M)} \bigvee_{i=1}^{g-1}S^2$. Moreover, he shows that $\pi_2(M)$ is a free $\mathbb{Z}[\pi_1(M)]$-module on $(g-1)$ generators. The generators correspond to some choice of distinct separating spheres in $M$; I like choosing $S_1, S_2, S_3$ pictured below (for genus 4). Note that the undrawn $S_4$ satisfies $-S_4 = S_1 + S_2 + S_3$ in $\pi_2(M)$.

    The question still makes sense if $M_i$-s are not aspherical, but in that case I do not know $\pi_2(M)$.

  2. In $\pi_2(M) // \Gamma$, the classes of the generators all coincide: $[S_1]= [S_2]= \cdots = [S_g]$. So, $g\cdot [S_1] = [S_1] + ... + [S_g] = 0$. For the same reason, if $n < g$, $n \cdot [S_1]$ has a representative which is an embedded separating sphere that "cuts off genus $n$ from $M$".

I should also mention that I do not understand how the $\pi_1(M)$-action on $\pi_2(M)$ relates to the $\Gamma$-action. I think that if $p \in \pi_1(M)$ does not intersect some embedded sphere representative of $s \in \pi_2(M)$, then the element $\gamma_p \in \Gamma$ that corresponds to "point-pushing" around $p$ would satisfy $\gamma_p \cdot s = p \cdot s$. (It helps that we can assume that $p$ does not intersect itself, since we're in dimension 3). If $p$ does intersect $s$, I do not understand what happens.

Picture of a 3-sphere with $g=4$ aspherical 3-manifolds attached and the separating 2-spheres:

Picture of separating spheres

(Everything in the picture of course has one less dimension than it should)

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  • $\begingroup$ Could you expand on point 2)? How do you know that $\left[S_1\right]=\ldots=\left[S_g\right]$? $\endgroup$ – ThiKu May 15 '14 at 23:17
  • $\begingroup$ @user39082: The mapping class group includes an element that rotates the picture I drew by 90 degrees clockwise. This takes $S_1$ to $S_2$, and thus identifies them in $\pi_2(M)//\Gamma$. $\endgroup$ – Ilya Grigoriev May 15 '14 at 23:38
  • $\begingroup$ So you assume that the $M_i$ are diffeomorphic? $\endgroup$ – ThiKu May 16 '14 at 1:05
  • $\begingroup$ @user39082 Yes, I do. That is a very good point, thank you. $\endgroup$ – Ilya Grigoriev May 16 '14 at 1:15
  • $\begingroup$ Your question is essentially answered by an old paper of Cesar de Sa, Rourke and Rourke. Hatcher has his version of the paper here: math.cornell.edu/~hatcher/Papers/DR3M.pdf The generators of the group are basically automorphisms of the factors, together with the automorphisms where one switches the factors. $\endgroup$ – Ryan Budney May 16 '14 at 2:32

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