Intersection forms of even 4-manifolds
Abstract
Abstract: In this talk, I will give a brief survey on some of the recent
development of 4-manifold theory and especially Furuta's proof of 10/8-
conjecture for spin 4-manifolds. Then I will outline a variation of
this
10/8 conjecture in my work with Tian-Jun Li. Roughly when a 4-manifold M
has no spin structure but has even intersection form $H_2(M,Z) = a E_8
\oplus b H $, then the number $b$ of the hyperbolic pairings is greater
than or equal to the number $a$ of $E_8$. We proved this under the
assumption that the 2-primary torsion part of $H_1(M,Z)$ is isomorphic
to
$Z_{2^i}$ for some i or $Z_2\oplus Z_2$. Independently Christian Bohr
obtained a more general result in terms of the fundamental group of
4-manifold. If time allows, I hope also to explain my recent work with
Christian Bohr on developing invariants of 3-manifold using Furuta's
result.