Andrew Blumberg, Floer homotopy theory and Morava K-theory

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  • #homotopy/stable-homotopy/equivariant #geomtop/Floer-theory
• Refs:
  • The paper: https://arxiv.org/pdf/2103.01507.pdf #resources/papers
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  • Floer homotopy

Andrew Blumberg, Floer homotopy theory and Morava K-theory

Reference: Andrew Blumberg, Floer homotopy theory and Morava K-theory. Princeton Algebraic Topology Seminar

Tags: #homotopy #geomtop/Floer-theory #projects/notes/seminars

• Some relevant things about Morava K-theory, which will be coefficients that we take.

• Tate construction and homotopy orbits and homotopy fixed points in the norm map,

• Produce a virtual fundamental class for flow categories. Moduli spaces of trajectories appearing will be “derived orbifold”.

  • Use some derived/spectral version of the usual Novikov ring?

• Flow category:

• Enriched in spaces

• Use a so-called Kuranishi flow category.

• Uses norm map

• Where do flow categories come from? One natural source: Morse functions. Objects are critical points, morphisms are roughly trajectories.

• Given a functor \(\mathsf{C} \to {\mathsf{Sp}}\) out of a flow category, can construct a spectrum as a \(\mathop{\mathrm{hocolim}}\). Turns Floer data into stable homotopy data?

• Defining a flow category:

• Morphisms are moduli spaces of Floer trajectories, i.e. flow lines using the symplectic form?

• Action map \(A: P\to {\mathbf{R}}\) given by integration, objects of \(P\) form a poset.

• Action induces a filtration \(P_{ \lambda} \coloneqq\left\{{ p\in P {~\mathrel{\Big\vert}~}A(p) < \lambda}\right\}\).

• Get filtered modules, need to work with a completion of this filtration. Tough to build/describe, uses point-set language and zigzags. Maybe easier in infinity category language?

• Get an \(E_2\) ring spectrum – not great! But there is an Ore condition that makes things nicer.

• Can express enrichments in terms of lax functor from an indexing 2-category

  • Cats of objects of set type \(S\) and morphism type \(V\) are lax functors from some indiscrete indexing 2-category (associated to \(S\)?) to a bicategory associated to \(V\).
  • Lax functors on bicategories: yikes.
  • A lot of homotopy coherence problems to deal with.

• How does one construct the virtual fundamental class? Under orientability hypothesis, have some kind of Alexander duality.

• See the Borel construction, the Adams isomorphism, ambidexterity.

  • Orientability hypothesis allows using the Thom isomorphism on equivariant cobordism. See complex stability.
  • Uses some model of iterated cones, similar to Khovanov stuff?

• Cover moduli space by Kuranishi charts \((X, G, s, Y)\)

  • symmetric monoidal structure is basically component-wise.
  • Morphisms are complicated, essentially involves transversality conditions (free actions, topological submersions, etc).
  • Stronger requirements than what John Pardon (?) imposes in his work on virtual fundamental classes.

• Technical issues with loop actions on categories, only get module structure on homotopy category?

  • Also issues with local-to-global coherence, need \(\mathop{\mathrm{hocolim}}\) to be compatible with Kuranishi atlases? And technical tools like Adams isomorphism and norm map. Need lots of equivariant stable homotopy theory.
  • Substantial difficulties extracting this data from the symplectic structure.
    • “This data”: stratified oriented Kuranishi structures?
  • Build a spectrum with some bar construction.

• Can get the Arnold conjecture from this homotopy type! I.e. \(\operatorname{rank}\operatorname{HF}_*(H, \Lambda) \leq {\sharp}\text{ orbits}\), here we’re taking homotopy groups. \(H\) is a Hamiltonian.

  • Can split \(H_*(M; \Lambda)\) off from \(\operatorname{HF}_*(H; \Lambda)\).

  • Recover statement about \(H{ \mathbf{F} }_p\) cohomology using the Atiyah Hirzebruch spectral sequence.

• Technical problems: virtual cochains are not functorial in morphisms of Kuranishi charts. No canonical map between certain cofibers. There are maps, they require choices, tracking them is tough.

  • Solution: degenerate to the normal cone to produce zigzags.

  • Sits between two Kuranishi charts by looking at the ends of the cone.

• A lot of cool stuff for homotopy theorists to do here!

  • See Cohen-Jones-Segal.

  • Homotopy theory should yield interesting symplectic consequences.

  • Work to build spectrally enriched Fukaya category.

  • Should be doing global homotopy theory or global homology, yielding a nicer way to describe all of this. Rezk is a proponent!

  • Can one do this over other spectra? Need strong Orientability of spectra conditions. Don’t expect a lift to the sphere spectrum, but maybe \({\operatorname{ku}}, \mu\).

• Improvement over rational results when there is torsion. May not improve over numerical bounds for special classes of MOCs/Symplectic geometry]].

• Morava K-theory is useful here because it behaves well with respect to finite groups

  • Want to dualize classifying space of orbifold, use Poincare duality for orbifolds. Equivariant stuff appears as an alternative to going to \(H_*(X; {\mathbf{Q}})\).

• Comment by Morava: the \(p=2\) case may be resolvable, see Boardman’s last paper. Set up some category of modules over the \(p=2\) Bockstein.