Suspension spectra and homology equivalences

Equivalences for Morse Homology

An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriately leads to a pseudo-cycle and a well-defined integral homology class in singu...

Moduli of Suspension Spectra

For a 1-connected spectrum E, we study the moduli space of suspension spectra which come equipped with a weak equivalence to E. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie’s calculus of homotopy fun...

Symmetric Spectra and Topological Hochschild Homology

A functor is defined which detects stable equivalences of symmetric spectra. As an application, the definition of topological Hochschild homology on symmetric ring spectra using the Hochschild complex is shown to agree with Bökstedt’s original ad hoc definition. In particular, this shows that Bökstedt’s definition is correct even for non-connective, non-convergent symmetric ring spectra. Mathem...

Suspension Spectra are Harmonic In memory

The notion of a harmonic spectrum in stable homotopy theory was introduced by the second author in [Rav84]. Roughly speaking, a spectrum is harmonic if its homotopy is computable with the chromatic spectral sequence. It was shown in [Rav84, Theorem 4.4] that a large class of connective spectra (including all finite spectra) are harmonic. On the other hand, the EilenbergMacLane spectrum for a to...

-homology of Commutative Rings and of E1 Ring Spectra