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

José Luis Cisneros-Molina and David Mond

Journal of Singularities
volume 24 (2022), 190-212

Received: 31 January 2022. Received in revised form: 20 October 2022.

DOI: 10.5427/jsing.2022.24h

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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, obtained by Gabrielov, Vorobjov and Zell, 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 revealing than earlier proofs.

2010 Mathematical Subject Classification:

55-02, 55T99, 32S05

Key words and phrases:

Multiple-point space, spectral sequence, alternating homology

Author(s) information:

José Luis Cisneros-Molina
Instituto de Matemáticas
Unidad Cuernavaca
Universidad Nacional Autónoma de México
Av. Universidad s/n, Col. Lomas de Chamilpa
Cuernavaca, Morelos, Mexico
email: jlcisneros@im.unam.mx

David Mond
Mathematics Institute
University of Warwick
Coventry CV4 7AL, U.K.
email: d.m.q.mond@warwick.ac.uk