--- date: 2022-02-23 18:45 modification date: Friday 25th March 2022 20:55:35 title: "Andrew Blumberg, Floer homotopy theory and Morava K-theory" aliases: [Andrew Blumberg, Floer homotopy theory and Morava K-theory] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #homotopy/stable-homotopy/equivariant #geomtop/Floer-theory - Refs: - The paper: #resources/papers - Links: - [Floer homotopy](Floer%20homotopy.md) --- ## 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](Morava%20K-theory), which will be coefficients that we take. - [Tate construction](Tate%20construction) and [homotopy orbits](homotopy%20orbits) and [homotopy fixed points](homotopy%20fixed%20points.md) in the [norm map](norm%20map.md), ![](attachments/image_2021-05-06-13-11-30.png) - Produce a [virtual fundamental class](virtual%20fundamental%20class.md) for [flow categories](flow%20categories.md). [Moduli spaces](Moduli%20spaces) of trajectories appearing will be "derived [orbifold](orbifold.md)". - Use some derived/spectral version of the usual [Novikov ring](Novikov%20ring.md)? - Flow category: ![](attachments/image_2021-05-06-13-15-11.png) - Enriched in spaces - Use a so-called [Kuranishi flow category](Kuranishi%20flow%20category). - Uses [norm map](norm%20map.md) - Where do flow categories come from? One natural source: Morse functions. Objects are critical points, morphisms are roughly trajectories. - Given a functor $\cat{C} \to \Sp$ out of a flow category, can construct a spectrum as a $\hocolim$. Turns Floer data into stable homotopy data? ![](attachments/image_2021-05-06-13-17-37.png) - Defining a flow category: ![](attachments/image_2021-05-06-13-18-44.png) - Morphisms are moduli spaces of Floer trajectories, i.e. flow lines using the symplectic form? - Action map $A: P\to \RR$ given by integration, objects of $P$ form a poset. - Action induces a filtration $P_{ \lambda} \da \ts{ p\in P \st A(p) < \lambda}$. - 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](infinity%20categories.md) language? - Get an $E_2$ [ring spectrum](ring%20spectrum) -- not great! But there is an Ore condition that makes things nicer. - Can express enrichments in terms of [lax functor](lax%20functor.md) 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](bicategory.md) associated to $V$. - Lax functors on bicategories: yikes. - A lot of [homotopy coherence](homotopy%20coherence.md) problems to deal with. - How does one construct the [virtual fundamental class](virtual%20fundamental%20class.md)? Under orientability hypothesis, have some kind of [Alexander duality](Alexander%20duality.md). ![](attachments/image_2021-05-06-13-32-51.png) - See the [Borel construction](Borel%20construction), the [Adams isomorphism](Adams%20isomorphism), [ambidexterity](ambidexterity). - Orientability hypothesis allows using the [Thom isomorphism](Thom%20isomorphism) on [equivariant cobordism](equivariant%20cobordism). See [complex stability](complex%20stability). - Uses some model of iterated cones, similar to [Khovanov](Khovanov) stuff? - Cover moduli space by [Kuranishi charts](Kuranishi%20charts) $(X, G, s, Y)$ - [symmetric monoidal](Symmetric%20monoidal%20category) 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](virtual%20fundamental%20classes). - Technical issues with loop actions on categories, only get module structure on homotopy category? - Also issues with local-to-global coherence, need $\hocolim$ to be compatible with Kuranishi atlases? And technical tools like [Adams isomorphism](Adams%20isomorphism) and [norm map](norm%20map.md). Need lots of [equivariant stable homotopy theory](equivariant%20stable%20homotopy%20theory.md). - Substantial difficulties extracting this data from the symplectic structure. - "This data": stratified oriented Kuranishi structures? - Build a spectrum with some [bar construction](bar%20construction.md). - Can get the [Arnold conjecture](Arnold%20conjecture) from this homotopy type! I.e. $\rank \HF_*(H, \Lambda) \leq \size\text{ orbits}$, here we're taking homotopy groups. $H$ is a [Hamiltonian](Hamiltonian.md). - Can split $H_*(M; \Lambda)$ off from $\HF_*(H; \Lambda)$. - Recover statement about $H\FF_p$ cohomology using the [Atiyah Hirzebruch spectral sequence](Atiyah%20Hirzebruch%20spectral%20sequence.md). - 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](normal%20cone) to produce [zigzags](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](Fukaya%20category.md). - Should be doing [global homotopy theory](global%20homotopy%20theory) 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](Orientability%20of%20spectra) conditions. Don't expect a lift to the [sphere spectrum](sphere%20spectrum), but maybe $\ku, \mu$. - Improvement over rational results when there is torsion. May not improve over numerical bounds for special classes of MOCs/Symplectic geometry]]. - [Morava K-theory](Morava%20K-theory) is useful here because it behaves well with respect to finite groups - Want to dualize [classifying space](classifying%20space.md) of [orbifold](orbifold.md), use [Poincare duality](Poincare%20duality) for orbifolds. Equivariant stuff appears as an alternative to going to $H_*(X; \QQ)$. - 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](Bockstein.md).