Recently, 3&4 manifolds with finite group actions has become a popular topic. Real manifolds form the simplest class among them. Following the strategy of Manolescu, Kronheimer-Mrowka, respectively, Konno- Miyazawa-Taniguchi and Li introduced two versions of real Seiberg-Witten-Floer homologies and developed many interesting applications. In this work, we show the two homology theories are equivalent whenever they are both defined following the strategy developed by Lidman and Manolescu. As application, we identify Froyshov-type invariants from two theories and proved some Smith-type inequalities.

In this talk, we first review the two approaches to (real) Seiberg-Witten-Floer homologies, then sketch a proof of our main theorem. If time admits, we talk about the applications and some basic examples.