Real Floer Homotopy Types and $w$-Invariants

Abstract

We express the local-equivalence classes of real Seiberg--Witten Floer homotopy types for spin Seifert $3$-manifolds with certain odd involutions in terms of the Fukumoto-Furuta $w$-invariant. The proof uses real orbifold Bauer-Furuta invariant for a class of spin $4$-orbifolds constructed by Fukumoto-Furuta-Ue. For torus knots, we prove that the local-equivalence classes of real Floer homotopy types associated with their $2^k$-fold cyclic branched covers are eventually periodic in $k$. Finally, we prove that the integer-valued concordance homomorphisms $\{8\delta_R^{(k)}\}_{k\geq1}$ are linearly independent.