Homological stability for spaces of embedded surfaces

Cantero, Federico¹ ; Randal-Williams, Oscar²

¹ Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron, 2, 1348 Louvain-la-Neuve, Belgium
² Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Geometry & topology, Tome 21 (2017) no. 3, pp. 1387-1467.

Résumé

We study the space of oriented genus-$g$ subsurfaces of a fixed manifold $M$ and, in particular, its homological properties. We construct a “scanning map” which compares this space to the space of sections of a certain fibre bundle over $M$ associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees.

Our results are analogous to McDuff’s theorem on configuration spaces, extended from $0$–dimensional submanifolds to $2$–dimensional submanifolds.

Détail

DOI : 10.2140/gt.2017.21.1387

Classification : 55R40, 57R20, 57R40, 57R50, 57S05
Keywords: submanifolds, characteristic classes, homology stability, embedding spaces, mapping class groups, scanning

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