On the E^1-term of the gravity spectral sequence

TL;DR

This paper proves that the E^1-term of a specific spectral sequence for loop space homology is isomorphic to the cobar construction, linking it to the classical Eilenberg-Moore spectral sequence.

Contribution

It establishes the isomorphism between the E^1-term of the gravity spectral sequence and the cobar construction, clarifying its relation to the classical spectral sequence.

Findings

E^1-term is isomorphic to the cobar construction

Spectral sequence aligns with the classical Eilenberg-Moore sequence

Applicable to variants of the spectral sequence

Abstract

The author constructed a spectral sequence strongly converging to h_*(Omega^n Sigma^n X) for any homology theory in [Topology 33 (1994) 631-662]. In this note, we prove that the E^1-term of the spectral sequence is isomorphic to the cobar construction, and hence the spectral sequence is isomorphic to the classical cobar-type Eilenberg-Moore spectral sequence based on the geometric cobar construction from the E^1-term. Similar arguments can be also applied to its variants constructed in [Contemp Math 293 (2002) 299-329].