A general Fredholm theory I: A splicing-based differential geometry

By Helmut Hofer, K. Wysocki, E. Zehnder. 2006

139 citations recorded by the source corpus.

This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. These spaces, in general, are locally not homeomorphic to open sets in Banach spaces. The present paper describes some of the differential geometry of this new class of spaces. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory, and Symplectic Field Theory

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7feba6e833f456e36f05e2fddd4d3b352e772ad5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A General Fredholm Theory II: Implicit Function Theorems (2007, background): “By σa we shall denote the collection of all indices i ∈ {1, . . . , n} for which ai = 0 and denote the complementary set of indices in {1, . . . , n} by σca.”
A general Fredholm theory III: Fredholm functors and polyfolds (2008, methodology): “11 in [25], the integer d does not depend on the choice of the M-polyfold chart used.”
Disk enumeration on the quintic 3-fold (2006, methodology): “We anticipate that with the completion of the generalized Fredholm theory currently being introduced in [8, 9], the ideas of Sections 1-5.”
On manifolds with corners (2009, background): “What geometric structure should we put on M g,h (M,L, J, β)? Hofer, Wysocki and Zehnder [4] make M l,m g,h (M,L, J, β) into a polyfold (actually, the zeroes of a Fredholm section of a polyfold bundle over a polyfold).”
Leaf-wise intersections and Rabinowitz Floer homology (2008, unknown): “In fact, since there are no automorphism groups acting, the framework of M -polyfolds [HWZ07, HWZ09] is suﬃcient to achieve transversality.”
Sc-Smoothness, Retractions and New Models for Smooth Spaces (2010, background): “In paper [9], the authors have described a generalization of differential geometry based on the notion of splicings.”
First steps in stable Hamiltonian topology (2010, methodology): “The assumption on transversality for holomorphic curves will be satisfied once the polyfold program of Hofer, Wysocki and Zehnder [31] is finished.”
Lecture Notes on Embedded Contact Homology (2013, background): “In SFT, it is expected that the polyfold theory of Hofer-Wysocki-Zehnder [20] will resolve these difficulties.”
Integration theory on the zero sets of polyfold Fredholm sections (2007, methodology): “1 in [7] that the splicing S = (π,E, V ) consists of the sc-Banach space E, the open subset V of a partial quadrant C in the sc-Banach space W , and the sc-smooth map π : V ⊕E → E has the property that for every v ∈ V the mapping πv = π(v, ·) : E → E is a bounded linear projection operator.”
Effect of Legendrian surgery (2009, methodology): “However for the purposes of this exposition we ignore this difficulty and claim that the correct version can be built according to the scheme presented here using one of the, for this purpose, currently developed technologies, notably, the polyfold theory of H Hofer, K Wysocki, and E Zehnder [31; 32”
Polyfold and Fredholm Theory (2014, background): “The initial version of the paper, [22], was written in 2008 on the occasion of a lecture given at the Clay Mathematical Institute (CMI), and described the theory as developed in [24, 25, 26].”
Framed bordism and Lagrangian embeddings of exotic spheres (2008, background): “We hope that the polyfold theory of [11], applied to this relatively simple setting, would give a clean way of circumventing this problem.”
Rational symplectic field theory over Z2 for exact Lagrangian cobordisms (2006, methodology): “Perturbation schemes to handle similar situations have been introduced by Fukaya, Oh, Ohta, and Ono [17] and by Hofer, Wysocki, and Zehnder [20].”