Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory

In 1995, Cohen, Jones and Segal proposed a method of upgrading any given Floer homology to a stable homotopy-valued invariant. For a generic pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously construct the conjectural stable normal framings, which are an essential ingredient in their construction, and give a rigorous proof that the resulting stable homotopy type recovers $\Sigma^\infty_+ M$. We further show that one can recover Thom spectra $M^E$ for all reduced $KO$-theory classes $E$ on $M$, by using slightly modified stable normal framings. Our paper also includes a construction of the smooth corner structure on compactified moduli spaces of broken flow lines with free endpoint, a formal construction of Piunikhin-Salamon-Schwarz type continuation maps, and a way to relax the stable normal framing condition to orientability in orthogonal spectra.