Two exact sequences for lattice cohomology

TL;DR

This paper proves two exact sequences for lattice cohomology associated with non-degenerate plumbing graphs, extending known results over integers and relating to Seiberg-Witten invariants.

Contribution

It establishes two new exact sequences for lattice cohomology over integers and introduces the concept of relative lattice cohomology linked to Seiberg-Witten invariants.

Findings

Proved the surgery exact sequence over integers for lattice cohomology.

Introduced relative lattice cohomology and computed its Euler characteristic.

Connected lattice cohomology with Seiberg-Witten invariants.

Abstract

We establish two exact sequences for the lattice cohomology associated with non-degenerate plumbing graphs. The first is the analogue of the surgery exact triangle proved by Ozsvath and Szabo for the Heegaard-Floer invariant HF^+; for the lattice cohomology over Z_2-coefficients it was proved by J. Greene. Here we prove it over the integers, and we supplement it by some additional properties valid for negative definite graphs. The second exact sequence is an adapted version which does not mix the classes of the characteristic elements (spin^c-structures); it was partially motivated by the surgery formula for the Seiberg-Witten invariant obtained by Braun and the author. For this we define the `relative lattice cohomology' and we also determine its Euler characteristic in terms of Seiberg-Witten invariants.