Multiple points of a simplicial map and image-computing spectral sequences

TL;DR

This paper presents new, simplified proofs for spectral sequences that compute the homology of images of finite maps, broadening applicability to many maps in singularity theory.

Contribution

Provides new, more canonical proofs for spectral sequences related to the homology of images of finite maps, applicable to a wide class of triangulable maps.

Findings

New proofs are conceptually simpler and more canonical.

Results apply to a broad class of maps, including most finite maps in singularity theory.

Spectral sequences effectively compute homology of images using multiple point spaces.

Abstract

The Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which computes the homology of the image of a closed map from the homology of $k$-fold fibred products of the map. We give new proofs of these results, in case the map can be triangulated. Thanks to work of Hardt, this holds for a very wide range of maps, and in particular for most of the finite maps of interest in singularity theory. The proof seems conceptually simpler and more canonical than earlier proofs.