Volume 19, issue 1 (2019)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 7, 3219–3760
Issue 6, 2687–3218
Issue 5, 2145–2685
Issue 4, 1601–2143
Issue 3, 1073–1600
Issue 2, 531–1072
Issue 1, 1–529

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
Constructing the virtual fundamental class of a Kuranishi atlas

Dusa McDuff

Algebraic & Geometric Topology 19 (2019) 151–238

Consider a space X, such as a compact space of J–holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of X by representing X via the zero set of a map SM: M E, where E is a finite-dimensional vector space and the domain M is an oriented, weighted branched topological manifold. Moreover, SM is equivariant under the action of the global isotropy group Γ on M and E. This tuple (M,E,Γ,SM) together with a homeomorphism from SM1(0)Γ to X forms a single finite-dimensional model (or chart) for X. The construction assumes only that the atlas satisfies a topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. However, if X is presented as the zero set of an sc–Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold M that uses an sc–smooth partition of unity.

virtual fundamental cycle, virtual fundamental class, pseudoholomorphic curve, Kuranishi structure, weighted branched manifold, polyfold
Mathematical Subject Classification 2010
Primary: 18B30, 53D35, 53D45, 57R17, 57R95
Received: 16 October 2017
Revised: 5 September 2018
Accepted: 15 September 2018
Published: 6 February 2019
Dusa McDuff
Mathematics Department
Barnard College
Columbia University
New York, NY
United States