• Tags
  • #homotopy/stable-homotopy
• Refs:
  • Strickland’s 2020 Intro to Spectra: https://arxiv.org/pdf/2001.08196.pdf#page=1
  • Synthetic spectra: https://math.mit.edu/~burklund/cookware-v0.2.pdf#page=4 #resources/notes
• Links:
  • structured ring spectrum
  • cohomology theory
  • How to extract homology using spectra
  • Homotopy groups of spectra
  • E-infty ring spectrum
  • t-structure
  • Moore spectrum
  • May recognition principle
  • Unsorted/cobordism spectrum
  • Unsorted/Dold-Kan correspondence

Spectra

Opinion from MO: modern Topology starts with spectra.

Important Facts

On existence of smash products:

Motivation

Suspension

We have another aim in constructing the category of spectra. In homology theory the suspension homomorphism \(\Sigma_*: h_n(X)\to h_{n+1}(\Sigma X)\) is always an isomorphism

Cohomology Theories

There are some things that spaces hide to cohomology theories, and we would like to mod out by this “extra information” that we don’t really need when we study spaces by means of cohomology theories.

The information that spaces hide is the unstable phenomenon, in the following sense : if \(X\) and \(Y\) are stably equivalent, for example \(\Sigma X \simeq\Sigma Y\), then \begin{align*} E_*(X)\cong E_{*+1}(\Sigma X) \cong E_{*+1}(\Sigma Y)\cong E_*(Y) \end{align*} for any cohomolology theory \(E_*\). This says that there is no cohomology theory that is going to see a difference between \(X\) and \(Y\), so we might as well says that they are “the same”.

Cohomology theories do not distinguish stably equivalent spaces.

Representability

Another good consequence of spectra is the Brown representability theorem. It says that any generalized cohomolology theory on spaces is representable by a spectra.

Hopf Invariant 1

A reason to care about cohomology theories: Adams’ two solutions to the Hopf invariant one problem.

His second proof (with Atiyah) is beautiful and short, but only because he uses an extraordinary cohomology theory, complex K-theory.

Categorical Properties

You may be familiar with a similar problem at the space level. One can construct the sequential colimit**.

So you can work with spaces and maps-up-to-homotopy if you like, but you won’t be able to do much. It’s much better to work with spaces and maps on-the-nose, and to make constructions like the pushout or sequential colimit.

Slogan: Pass to homotopy as late as possible.

Noting that \(\Sigma S^n = S^{n+1}\), we could alternatively define \(\mathbb{S} \coloneqq\lim_k \Sigma^k S^0\), then it turns out that \(\pi_n \mathbb{S} = \pi_n^S\).

This object is a spectrum, which vaguely resembles a chain complex with a differential: \begin{align*} X_0 \xrightarrow{\Sigma} X_2 \xrightarrow{\Sigma} X_3 \xrightarrow{\Sigma} \cdots \end{align*}

Spectra represent invariant theories (like cohomology) in a precise way. For example, \begin{align*} HG := {\left[ { K(G, 1), K(G, 2), \cdots } \right]} \end{align*}

then \(H^n(X; G) \cong [X, K(G, 1)]\), and we can similarly extract \(H^*(X; G)\) from (roughly) \(\pi_* HG := [\mathbb{S}, HG \smash X]\).

Note: this glosses over some important details! Also, smash product basically just looks like the tensor product in the category of spectra.

A modern direction is cooking up spectra that represent extraordinary cohomology theories. There are Eilenberg Steenrod axioms that uniquely characterize homology on spaces.

Homotopy: \(\pi_* E \coloneqq[{\mathbb{S}}, E]_*\) Cohomology: \(H^*(X; M) = [\mathop{\mathrm{{\Sigma_+^\infty}}}X, HM]_{-*}\). Homology: \(H_*(X; M) = [{\mathbb{S}}, HM \wedge X]_*\).

Spectrum \(E\) with coefficients in \(G\): \(E \wedge MG\) for \(MG\) the corresponding Moore spectrum.

Spectra

• On notation: capital letters denote some usual spectrum, eg \({{\mathbf{B}}{\operatorname{O}}}\) or \({\operatorname{KU}}\), while lowercase denotes their connective covers, e.g. \(\bo\) or \({\operatorname{ku}}\).
  • Connectivity: a spectrum \(X\) is \(i{\hbox{-}}\)connected if \(\pi_{<i} X =0\). Use inclusion of subcategory \({\mathsf{Sp}}_{<i} \hookrightarrow{\mathsf{Sp}}\) to produce a right-adjoint \(\tau_{\geq i}\), the \(i{\hbox{-}}\)connective cover.