Intersection forms of 4-manifolds with boundary - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:27:21Z http://mathoverflow.net/feeds/question/102068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102068/intersection-forms-of-4-manifolds-with-boundary Intersection forms of 4-manifolds with boundary david-sun 2012-07-12T20:23:37Z 2012-07-20T22:06:26Z <p>Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a fixed integer $k \neq 0, \pm 1$. Is it true that the number of possible intersection forms for such $X$ is finite? Any reference would be appreciated. </p> http://mathoverflow.net/questions/102068/intersection-forms-of-4-manifolds-with-boundary/102073#102073 Answer by Daniele Zuddas for Intersection forms of 4-manifolds with boundary Daniele Zuddas 2012-07-12T21:14:41Z 2012-07-12T21:14:41Z <p>As the question is posed, the answer is not. If you take connected sum with $\Bbb{CP}^2$ you preserve the determinant, but change the intersection form.</p> http://mathoverflow.net/questions/102068/intersection-forms-of-4-manifolds-with-boundary/102776#102776 Answer by Agol for Intersection forms of 4-manifolds with boundary Agol 2012-07-20T22:06:26Z 2012-07-20T22:06:26Z <p>The classification of integral quadratic forms is discussed in <a href="http://books.google.com/books?id=upYwZ6cQumoC&amp;lpg=PR5&amp;ots=_L-R4Qi9DS&amp;dq=conway%2520sloane&amp;lr&amp;pg=PA352#v=onepage&amp;q&amp;f=false" rel="nofollow">Chapter 15 of Conway-Sloane</a>. In particular, the discussion there implies that there are only finitely many integral quadratic forms of a given determinant and dimension. Section 11 in the chapter discusses methods for computing the number of such forms. </p>