On the Spectral Sequence of the Swiss-cheese Operad

TL;DR

This paper proves the Swiss-cheese operad's homology is Koszul, showing the associated spectral sequence collapses early, but the resulting operad differs from the original homology, confirming a conjecture by Voronov.

Contribution

It establishes the Koszul property of the Swiss-cheese operad's homology and clarifies the spectral sequence's behavior and limitations.

Findings

Spectral sequence collapses at the second stage.

The second-stage operad differs from the homology.

Confirmed Voronov's conjecture on spectral sequence collapse.

Abstract

We prove that the homology of the Swiss-cheese operad is a Koszul operad. As a consequence, we obtain that the spectral sequence associated to the stratification of the compactification of points on the upper half plane collapses at the second stage, proving a conjecture by A. Voronov. However, we prove that the operad obtained at the second stage differs from the homology of the Swiss-cheese operad.