Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory
Ciprian Mircea Bonciocat
TL;DR
This work provides a rigorous Morse–Bott realization of the Cohen–Jones–Segal construction, showing that with appropriate stable normal framings (including $E$-twisted framings for KO-theory twists), the CJS construction on the Morse–Bott flow category recovers Thom spectra $M^E$, and in particular $Σ^ obreak o M$ when $E=0$. It develops smooth corner structures for compactified moduli spaces, constructs PSS-type continuation maps, and demonstrates homotopy invariance and functoriality of the CJS framework. The paper further shows any stable normal framing stabilizes to an $E$-twisted framing, making Thom spectra the only stable homotopy types arising from CJS in this setting, and it extends the formalism to spectrum-valued orientations for greater flexibility. These results lay groundwork for Floer homotopy theory of monotone Lagrangians, including potential infinite-dimensional extensions and broader spectrum-oriented formulations via orthogonal spectra.
Abstract
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 alleged stable normal framings, which are an essential ingredient in their construction, and give a rigorous proof that the resulting stable homotopy type recovers $Σ^\infty_+ M$. We further show that other systems of compatible stable normal framings recover Thom spectra $M^E$, for all reduced $KO$-theory classes $E$ on $M$. 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.