Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A splitting theorem for the Seiberg-Witten invariant of a homology $S^1
  \times S^3$   We study the Seiberg-Witten invariant $\lambda_{\rm{SW}} (X)$ of smooth spin
$4$-manifolds $X$ with integral homology of $S^1\times S^3$ defined by Mrowka,
Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an
index-theoretic correction term. We prove a splitting formula for this
invariant in terms of the Fr{\o}yshov invariant $h(X)$ and a certain Lefschetz
number in the reduced monopole Floer homology of Kronheimer and Mrowka. We
apply this formula to obstruct existence of metrics of positive scalar
curvature on certain 4-manifolds, and to exhibit new classes of integral
homology $3$-spheres of Rohlin invariant one which have infinite order in the
homology cobordism group.

35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn