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Every closed connected oriented $4$-manifold has a signature, defined via a cohomological intersection form. In Turaev's book Quantum Invariants of Knots and 3-Manifolds the definition of a certain invariant of $3$-manifolds in chapter II uses the signature of a $4$-manifold which is not closed (actually it arises from attaching $2$-handles to a $4$-ball), which is defined using a homological intersection form.

Q1) How is this homological intersection form defined?

Q2) Why is the signature of $S^2 \times D^2$ zero? (Actually I don't know if this is true, but this seems to be used in a calculation in Turaev's book.)

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  • $\begingroup$ That's not a closed 4- manifold! $\endgroup$ – Cheerful Parsnip Jun 30 '14 at 20:43
  • $\begingroup$ Closed ordinarily means empty boundary in this context. $\endgroup$ – Ted Shifrin Jun 30 '14 at 20:44
  • $\begingroup$ I've replaced "closed" by "connected" :) $\endgroup$ – Martin Brandenburg Jun 30 '14 at 20:45
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    $\begingroup$ The signature can be thought of as the signature of the intersection form $H_2(M)\otimes H_2(M)\to \mathbb Z$. This may be the way it is defined for non-closed $4$-manifolds. In that case, it's easy to see that $H_2(M)$ is generated by $S^2\times \{0\}$, and this has trivial self-intersection since there is a parallel push-off which is disjoint, implying that the intersection form is identically $0$. $\endgroup$ – Cheerful Parsnip Jun 30 '14 at 21:14
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    $\begingroup$ This is not a comment, it is an answer ... but could you please also explain what the homological intersection form is? $\endgroup$ – Martin Brandenburg Jun 30 '14 at 21:27
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The signature can be thought of as the signature of the intersection form $H_2(M)\otimes H_2(M)\to\mathbb Z.$ This may be the way it is defined for non-closed 4-manifolds. In that case, it's easy to see that $H_2(M)$ is generated by $S^2×\{0\}$, and this has trivial self-intersection since there is a parallel push-off which is disjoint, implying that the intersection form is identically $0$.

The intersection form $H_2(M)\otimes H_2(M)\to \mathbb Z$ is defined by realizing $2$-dimensional homology classes as maps of surfaces into $M$. The algebraic intersection number between two surfaces is defined by first putting them in general position with respect to each other, and then counting their intersections with sign. Note that a surface in a $4$-manifold generically hits another surface in a finite collection of points.

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  • $\begingroup$ Thanks a lot! It's (almost) clear to me now. Only one question is left, why does every homology class arise from an embedded surface? $\endgroup$ – Martin Brandenburg Jul 1 '14 at 7:01
  • $\begingroup$ @MartinBrandenburg: See mathoverflow.net/questions/1489/…. I wasn't assuming that the surfaces are embedded, but you can assume that with no loss of generality. $\endgroup$ – Cheerful Parsnip Jul 1 '14 at 7:13
  • $\begingroup$ Thank you for the reference. $\endgroup$ – Martin Brandenburg Jul 1 '14 at 7:15

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