The equivalence between two real Seiberg-Witten Floer homologies

Abstract

We show that for a real rational homology sphere $Y$ equipped with a real $\mathrm{spi}{\mathrm{n}}^{\mathrm{c}}$ structure $\mathfrak{s}$, the real monopole Floer homology defined by Li and the real Seiberg-Witten Floer homology defined by Konno, Miyazawa and Taniguchi are isomorphic. As corollaries, we identify some Froyshov-type invariants and prove two Smith-type inequalities.

v2: 52 pages, no figure. Comparing to v1, minor changes throughout and corrections in Section 2.2.4 and proof of Theorem 1.5. Accepted by Advances in Mathematics