Floer theory and its topological applications

Manolescu, Ciprian

doi:10.1007/s11537-015-1487-8

Cited by 5 publications

(6 citation statements)

References 100 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a thorough introduction to the above Floer-theoretic invariants, connections between them, and their applications see [28], and references there. See also more recent papers [6], [14], [15] on symplectic instanton Floer homology, and [7] on Atiyah-Floer conjecture.…”

mentioning

confidence: 99%

Bordered theory for pillowcase homology

Kotelskiy

2019

Mathematical Research Letters

View full text Add to dashboard Cite

We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A∞ module M (L) over A. Then we prove that Lagrangian Floer homology HF (L, L ) is isomorphic to a suitable algebraic pairing of modules M (L) and M (L ). This extends the pillowcase homology construction given a 2-stranded tangle inside a 3-ball, if one obtains an immersed unobstructed Lagrangian inside the pillowcase, one can further associate an A∞ module to that Lagrangian.

show abstract

mentioning

confidence: 99%

Bordered theory for pillowcase homology

Kotelskiy

2019

Mathematical Research Letters

View full text Add to dashboard Cite

show abstract

“…In this section the author is relying heavily on the expository article by Manolescu [39]. We refer the reader to this beautiful survey of recent topological applications of Floer theory.…”

Section: Manolescu's Equivariant Floer Homotopy and The Triangulationmentioning

confidence: 99%

Floer homotopy theory, revisited

Cohen¹

2020

Handbook of Homotopy Theory

View full text Add to dashboard Cite

In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory" [12]. The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is the Floer homology isomorphic to the (singular) homology of a naturally occuring (pro)spectrum defined from the properties of the moduli spaces inherent in the Floer theory?". A proposal for how to construct such a spectrum was given in terms of a "framed flow category", and some rather simple examples were described. Years passed before this notion found some genuine applications to symplectic geometry and low dimensional topology. However in recent years several striking applications have been found, and the theory has been developed on a much deeper level. Here we summarize some of these exciting developments, and describe some of the new techniques that were introduced. Throughout we try to point out that this area is a very fertile ground at the interface of homotopy theory, symplectic geometry, and low dimensional topology.

show abstract

“…And recently, Manolescu [133] gave a spectacular application to higher dimensions, disproving the triangulation conjecture. His survey [132] gives an excellent description of the Floer homologies for 3-manifolds and of their topological applications. We therefore simply state a few results, that give an idea which kind of topological applications Floer homologies have.…”

Section: Applications To Topologymentioning

confidence: 99%

“…This is also the case in higher dimensions: Theorem 9.4. ( [132]) For every n 5 there exist compact n-dimensional topological manifolds that do not admit a triangulation.…”

Section: Applications To Topologymentioning

confidence: 99%

“…Other Floer homologies have applications to the structure of manifolds, like Property P for knots, the question which compact 3-manifolds can be obtained by Dehn surgery of S 3 on which knot, and the existence of topological manifolds of dimension 5 that admit no triangulation. Given Manolescu's excellent survey [132], we just state a few of these results in Section 9.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Floer Homologies, with Applications

Abbondandolo

Schlenk

2018

Jahresber. Dtsch. Math. Ver.

View full text Add to dashboard Cite

Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer's seminal ideas. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulisable. This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry. 14 4.

show abstract

Floer theory and its topological applications

Cited by 5 publications

References 100 publications

Bordered theory for pillowcase homology

Bordered theory for pillowcase homology

Floer homotopy theory, revisited

Floer Homologies, with Applications

Contact Info

Product

Resources

About