On the intersection form of surfaces (Q2436628)

The intersection form \(\operatorname {Int}\) of a closed, oriented surface \(M\) is a nondegenerate, skew symmetric, bilinear form on the first homology group \(H_1(M,R)\) which measures the algebraic intersection number of two homology classes. When \(M\) is provided with a Riemannian metric \(g\), the metric induces a norm \(\| \cdot\| _s\), called the stable norm, on \(H_1(M,R)\) which measures the length of the homology classes. One defines the norm of the intersection form relative to the stable norm through the formula \[ K(M,g) = \sup \left\{ {\operatorname{Int}(h_1,h_2) \over \| h_1\|_s\| h_2\|_s } : h_1,h_2 \in H_1(M,R)\backslash\{0\}\right\}. \] In this paper, the authors find upper and lower bounds on \(K(M,g)\) in terms of other invariants of the topology and geometry of \((M,g)\). They find bounds involving the diameter, 1-dimensional systole, and area of \((M,g)\) for general metrics, and bounds involving the systole and the genus for hyperbolic metrics.