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

@article{Hofer2006AGF, title={A general Fredholm theory I: A splicing-based differential geometry}, author={Helmut H. Hofer and Kris Wysocki and Eduard Zehnder}, journal={Journal of the European Mathematical Society}, year={2006}, volume={9}, pages={841-876} }

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

#### 119 Citations

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#### References

SHOWING 1-10 OF 37 REFERENCES

A General Fredholm Theory II: Implicit Function Theorems

- Mathematics
- 2007

Abstract.This is the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. In general, these spaces are not locally homeomorphic to open… Expand

A general Fredholm theory III: Fredholm functors and polyfolds

- Mathematics
- 2008

This is the third in a series of papers devoted to a general Fredholm theory in a new class of spaces, called polyfolds. We first introduce ep–groupoids and polyfolds. Then we generalize the Fredholm… Expand

Compactness results in Symplectic Field Theory

- Mathematics
- 2003

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic… Expand

Introduction to Symplectic Field Theory

- Mathematics, Physics
- 2000

We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds… Expand

NON-LINEAR FREDHOLM MAPS AND THE LERAY-SCHAUDER THEORY

- Mathematics
- 1977

ContentsIntroduction § 1. Banach manifolds and their maps § 2. The degree of a Fredholm map § 3. Equivariant Fredholm maps § 4. Solubility of equations with Fredholm operators § 5. Some applications… Expand

A General Fredholm Theory and Applications

- Mathematics
- 2004

The theory described here results from an attempt to find a general abstract framework in which various theories, like Gromov-Witten Theory (GW), Floer Theory (FT), Contact Homology (CH) and more… Expand

Fundamentals of differential geometry

- Mathematics
- 1998

This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance,… Expand

Introduction to Differentiable Manifolds

- Mathematics
- 1964

Foreword.- Acknowledgments.- Differential Calculus.- Manifolds.- Vector Bundles.- Vector Fields and Differential Equations.- Operations on Vector Fields and Differential Forms.- The Theorem of… Expand

Introduction to Symplectic Topology

- Mathematics
- 1995

Introduction I. FOUNDATIONS 1. From classical to modern 2. Linear symplectic geometry 3. Symplectic manifolds 4. Almost complex structures II. SYMPLECTIC MANIFOLDS 5. Symplectic group actions 6.… Expand

ON COBORDISM OF MANIFOLDS WITH CORNERS

- Mathematics
- 2000

This work sets up a cobordism theory for manifolds with corners and gives an identication with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of… Expand