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There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$: $$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# \overline{\mathbb{C}\mathbb{P}^2} \right) \right) \cong \left(\mathop{\#}^{m+b^+_2(M)} \mathbb{C}\mathbb{P}^2 \right) \# \left(\mathop{\#}^{m+b^-_2(M)} \overline{\mathbb{C}\mathbb{P}^2} \right)$$ The $b^\pm_2(M)$ are the positive and negative eigenvalues of the intersection form. This has far-reaching consequences, for example any 4-manifold invariant that is multiplicative under direct sum $\#$ will only measure the intersection form on simply connected manifolds.

Is there a similar statement for non-simply-connected 4-manifolds? What is the closest we can get?

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    $\begingroup$ Given that blow up/downs won't change the fundamental group, I'm not sure what kind of answer you're hoping for. $\endgroup$ – Donu Arapura Sep 12 '14 at 17:17
  • $\begingroup$ @DonuArapura, Ideally, a connected sum of $S^1 \times S^3$, $\mathbb{CP}^2$ and $\overline{\mathbb{CP}^2}$, but that's probably to much to expect. $\endgroup$ – Manuel Bärenz Sep 12 '14 at 17:19
  • $\begingroup$ But the fundamental group would have to be free for that to work, wouldn't it? $\endgroup$ – Donu Arapura Sep 12 '14 at 17:23
  • $\begingroup$ @DonuArapura, right. So an interesting, more realistic, similar statement could be that such a sum is possible for free $\pi_1$. Or maybe it would be possible to have a connected sum with $\mathbb{CP}^2$'s and a manifold with trivial intersection form. I know, maybe it's a soft question, but maybe there is a sensible generalisation. $\endgroup$ – Manuel Bärenz Sep 12 '14 at 17:27
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    $\begingroup$ Wall's theorem tells you that if $X$ and $X'$ are $h$-cobordant 4-manifolds, then $X\#k(S^2\times S^2)$ and $X'\#k(S^2\times S^2)$ are diffeomorphic for some $k$ (conjecturally, $k=1$ is enough). This is somehow related (just blow up once and you get a similar statement). Somehow it looks like you're looking for a "minimal model" of a 4-manifold -- much like in the minimal model program in algebraic geometry. $\endgroup$ – Marco Golla Sep 12 '14 at 20:14
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Two smooth oriented 4-manifolds $M,N$ with fundamental group $\pi$ admit maps to $B\pi$, the classifying maps of their universal covers. The manifolds $M$ and $N$ become diffeomorphic after connect summing with copies of complex projective space if and only if the images of the fundamental classes $[M],[N]$ in $H_4(B\pi)$ under the classifying maps are equal, for some choice of maps to $B\pi$, i.e. if we mod out $H_4(B\pi)$ by the action of outer automorphisms of $\pi$. This follows from Kreck's surgery theory with normal 1-type $B\pi \times BSO$. This theory really tells us about connect summing with $S^2 \times S^2$s, but $\#^2(\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}) \cong S^2 \times S^2 \# \mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. It turns out that the class in $H_4$ is determined by the stable isomorphism classes of the $\mathbb{Z}\pi$ modules $\pi_2(M)$, $\pi_2(N)$.

Thus for general groups you need many minimal models. For groups with vanishing $H_4$ the situation is rather similar to the trivial group situation; you just need to construct some 4-manifold with that group as its fundamental group, for the base manifold. (Analogous to a hidden $S^4$ on the right hand side of your equation.)

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  • $\begingroup$ Wow, that's really interesting. I'm actually constructing a TQFT that can be twisted by a 4-cocycle of a group, so thanks! $\endgroup$ – Manuel Bärenz Oct 27 '15 at 19:24

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