nLab polyfold

Idea

The notion of polyfolds, due to Helmut Hofer, is a notion of generalized smooth spaces. The bulk of the theory is a generalized approach to Fredholm operator analysis in geometrical situations; which can be considered as an analytic framework designed to systematically resolve the issue of transversality. It gives a unified framework for symplectic field theory, Floer homology, and some constructions in contact geometry etc.

References

Reviews

• Katrin Wehrheim‘s quick introduction.

• Joel W. Fish, Helmut Hofer, Lectures on Polyfolds and Symplectic Field Theory (arXiv:1808.07147)

Original articles

• Helmut Hofer, A general Fredholm theory and applications (arXiv:math/0509366)

• Helmut Hofer, Kris Wysocki, Eduard Zehnder,

  • A general Fredholm theory I: A splicing-based differential geometry (arXiv/0612604)

  • A general Fredholm theory II: implicit function theorems (arXiv:0705.1310)

  • A general Fredholm theory III: Fredholm functors and polyfolds (arXiv:0810.0736)

• Helmut Hofer, Kris Wysocki, Eduard Zehnder, Integration theory for zero sets of polyfold Fredholm sections (arXiv/0711.0781)

• Helmut Hofer, Polyfolds and a general Fredholm theory (arXiv/0809.3753)

• Helmut Hofer, Polyfolds and Fredholm Theory, Oxford University Press (2017) (doi:10.1093/oso/9780198784913.003.0004)

We thank Eugene Lerman for some of the information here.