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Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$ denote the connected sum of $M$ and $N$ along $S$ (see Wikipedia). I wish to study necessary conditions for $M\#_S N$ to be aspherical!

In the case of usual connected sum $M \#N$, we have that $M \# N$ is not aspherical if $M$ is not homotopic to $S^n$. In the above general connected sum $M\#_S N$, we have that $M\#_S N$ is not aspherical if $\pi_1(M\#_S N)$ is an amalgamated free product or an HNN extension over a subgroup of cohomological dimension less than $n-1$. But what happens if $\pi_1(M\#_S N)$ is neither? Are there any other sufficient conditions for $M\#_S N$ to be not aspherical?

I am considering the following simple example in dimension $4$. Let $M = S^2 \times T^2$ and $N = T^4$, where $T^n$ denotes the $n$-torus. Let $S = T^2$ and form $M\#_S N$ with $\pi_1$-injective inclusions. I think $\pi_1(M\#_S N) = \mathbb{Z}^4$ (the fundamental group of the $4$-torus). How do I decide whether $M\#_S N$ is aspherical in my case? If $M\#_S N$ were aspherical, then $M\#_S N = T^4$. Can I eliminate this possibility?

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  • $\begingroup$ I think there is an assumption in the 'usual connnected sum'-case missing. The connected sum of two orintable surfaces of geni $m,n$ is an orientable surface of genus $m+n$ and hence aspherial. $\endgroup$
    – HenrikRüping
    Commented Jul 19 at 15:19

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The fiber sum you describe is $T^4$. For you are removing $D^2 \times T^2$ from each of $M$ and $N$. But $M - D^2 \times T^2 = D^2 \times T^2$ so you're just removing $D^2 \times T^2$ from $N$ and putting it back in.

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  • $\begingroup$ I understand, thank you. However, if $M$ is a general $T^2$-bundle over $S^2$ that is not necessarily trivial and $S$ and $N$ are the same as above, then can something be said about the asphericity of $M\#_S N$, noting that perhaps $\pi_1(M\#_S N) = \mathbb{Z}^4$ again? $\endgroup$
    – Jeremy
    Commented Jul 18 at 4:50
  • $\begingroup$ It’s not hard to describe the universal cover of your manifold in terms of covers of $M-S$ and $N-S$; you can then try to compute the homology of the universal cover and decide if it’s contractible. See for instance the introductory parts of Cappell’s paper on the splitting theorem or papers of Plotnick and Suciu from the 80s. $\endgroup$
    – Danny Ruberman
    Commented Jul 18 at 14:02
  • $\begingroup$ If I'm not mistaken, Cappell’s paper discusses codimension $1$ case in the introduction, and the theorems are for dimensions $\ge 5$. I have $4$-manifolds with codimension $2$ submanifolds! $\endgroup$
    – Jeremy
    Commented Jul 19 at 0:13
  • $\begingroup$ I was referring to the description of the universal cover of the union. This is independent of dimension. $\endgroup$
    – Danny Ruberman
    Commented Jul 19 at 19:06

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