Takao Matumoto

A gap theorem for Lusternik–Schnirelmann π1-category

Abstract The Lusternik–Schnirelmann π 1 -category, cat π 1 X , of a topological space X is the least integer n such that X can be covered by n+1 open subsets U 0 ,…,U n , every loop in each of which is contractible in X . In this paper we will prove a gap theorem that cat π 1 M n ≠n−1 for any closed connected n -dimensional manifold M n . With the fact that the fundamental group of a compact Kahler manifold is not a nontrivial free group, we see as a corollary that the π 1 -category of a compact Kahler surface is even.

HYPERBOLIC METRIC ON THE MODULI SPACE OF BPST-INSTANTONS OVER S4

Publisher Summary This chapter discusses hyperbolic metric on the moduli space of BPST-instantons over S 4 . The moduli space m of 1-instantons over 4-sphere S 4 , with gauge group SU(2), is the homogeneous hyperbolic space. The chapter describes a natural hyperbolic metric over m by using the L 2 -norm of the 2-forms, which has a negative constant sectional curvature -5/32π 2 independent of the conformal deformation of S 4 . The stereographic projection of S 4 -{North pole} onto R 4 induces a conformally flat metric. In the case of a general Riemannian 4-manifold, the metric tensor gives only a semi-metric on the nonsingular part of the moduli space. However, on a simply connected 4-manifold with positive definite intersection form, it gives a Riemannian metric near the boundary of the moduli space of 1-instantons.