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- Tags: - #todo/untagged - Refs: - Cohen overview of Floer homotopy types - ✨ Crash course in homotopy with some words about infty-categories. - Links: - #todo/create-links

Homotopy talk

Linked topics to look at:

• Freudenthal suspension theorem
• monopole homotopy type

Results

• Furuta: \({\operatorname{Pin}}_2{\hbox{-}}\)equivariant Seiberg-Witten Floer \({\mathsf{K}}\) theory used to prove the Furuta 10 8 conjecture. See monopole homotopy type.
• Kneser 1928: does every topological manifold admit a PL structure (triangulation)? Resolved by Manolescu using the equivariant Seiberg-Witten homotopy type.
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  • Motivations:
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    • Singular homology:

Classical

• Definition of \(\pi_k X\)

• Suspension, smash product, based loop space

• Stabilization and Freudenthal suspension

• Fibrant and cofibrant resolution, weakly equivalent replacement by CW complexes

• Homotopy equivalence vs weak equivalence

• Big theorems:

  • Freudenthal suspension
  • Whitehead
  • Hurewicz
  • 

Compare to chain complexes:

Monoidal structure:

Homotopy groups:

Stable homotopy

• (Stable) homotopy types
• SHC as an infinity category, triangulated, monoidal
• Various path/loop spaces: \({\mathcal{L}}M, {\mathcal{P}}M, {\Omega}M, \Sigma M, A\wedge M\)
• The \({\Sigma}^\infty{\hbox{-}}{\Omega}^\infty\) adjunction, delooping
• Definition of a suspension and omega spectra,
• Pro spectra
• Homotopy groups of spectra
  • Colimits
• Wedges of spectra (as sums)
• Generalized cohomology theories: cobordism and \({\mathsf{K}}\)
  • The Eilenberg-Maclane spectra \(HR\)
  • Recovering homology as \(H^*(X; {\mathbf{Z}}) = \pi_* \qty{ {\Sigma}^\infty X \wedge{H{\mathbf{Z}}}}\)???
• Fibres/cofibres and their sequences
• Serre Sseq vs AdSseq
• ✨Ring spectra, \({\mathbb{E}}_\infty\) rings, and \(A_\infty\) structures
  • As monoid and commutative monoid objects in SHC
  • Can take THH of a ring spectrum to define invariants.
  • Allow cohomology operations generalizing the Steenrod operations.
  • R-module spectra
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• ✨Morava K theory
  • Why care: they’re like “fields”; every \(K(n){\hbox{-}}\)module \(M\) is free, i.e. \(M\cong \bigvee_i { \Sigma^{\scriptstyle[k_i]} K(n) }\) (a wedge of shifts of the original).
• The Pontryagin-Thom construction
• \({\operatorname{MSO}}\)
• Connective spectra
• Examples of usefulness:
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• Model categories, weak equivalence, Quillen equivalence

Equivariant homotopy

• Define equivariance
• Borel construction
• \(G{\hbox{-}}\)equivariant singular homology.
• Homotopy orbits and fixed points

Infty Cats

• Groupoids, \(\Pi_1 X\)
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• Simplicial sets
  • Realization/singular simplex adjunction
  • Kan complexes
  • The actual definition of an infty category;
• Classifying categories \({\mathbf{B}}\mathsf{C}\), every groupoid is equivalent to \({\textstyle\coprod}_{\alpha \in J}{{\mathbf{B}}G}_\alpha\) for some groups \(G_\alpha\).
• Colimits, pullbacks
• Geometric realization/nerve adjunction
• \(\mathop{\mathrm{Maps}}_{\mathsf{C}} = \colim(\Delta^0 \to \mathsf{C}{ {}^{ \scriptscriptstyle\times^{2} } } \leftarrow[\Delta^1, \mathsf{C}])\)
• Pro and Ind objects
• Symmetric monoidal categories
• Weak equivalences
• Contractible choices:
• Limits and colimits

Floer

• The Arnold conjecture
• Topological categories
• Flow categories
• The action functional on loop spaces \(f\mapsto \int_{{\mathbb{B}}^2} f^*\omega\) mapping \({\mathcal{L}}M\to {\mathbf{R}}\).
• Paths as cylinders: \([{\mathbf{R}}, {\mathcal{L}}M] \cong [{\mathbf{R}}\times S^1, M]\).
• The Floer homotopy theorem: