# Signature of $S^2 \times D^2$

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.)

• That's not a closed 4- manifold! – Cheerful Parsnip Jun 30 '14 at 20:43
• Closed ordinarily means empty boundary in this context. – Ted Shifrin Jun 30 '14 at 20:44
• I've replaced "closed" by "connected" :) – Martin Brandenburg Jun 30 '14 at 20:45
• 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 $0$. – Cheerful Parsnip Jun 30 '14 at 21:14
• This is not a comment, it is an answer ... but could you please also explain what the homological intersection form is? – Martin Brandenburg Jun 30 '14 at 21:27

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.