A guide for computing stable homotopy groups

Tags: #homotopy/stable-homotopy #resources/papers #projects/notes/reading Refs: stable homotopy

A Guide for Computing Stable Homotopy Groups

Refs

Paper: A Guide for Computing Stable Homotopy Groups

Notes

  • Main idea: certain types of “topological field theories” are classified by certain stable homotopy classes of maps between 2 specific spectra

  • Main tool: Adams spectral sequence, invented to resolve Hopf invariant one. Relates cohomology of spaces/spectra to stable homotopy

  • Mod-2 cohomology of any space is a module over the Steenrod algebra

  • Two important classes of isomorphisms: chain homotopy equivalences vs quasi-isomorphisms.

    • Derived category inverts quasi-isomorphisms (not an equivalence relation!)
    • Homotopy category inverts chain homotopy equivalences. In theory, easier to do.
    • For bounded below projective chain complexes, quasi-iso implies chain homotopy equivalence, so can take derived category to be projective chain complexes with chain homotopy equivalences as morphisms

  • Analogy:

    • Chain homotopy equivalences for modules \(\iff\) homotopy equivalences
    • Quasi-isomorphisms \(\iff\) weak homotopy equivalences
    • Derived category \(\iff\) homotopy category of spectra
    • Projective chain complexes \(\iff\) CW spectra

  • Standard examples of spectra:

    • Suspension spectrum: for any space \(X\), \begin{align*}\Sigma^\infty X \coloneqq\qty{ X_0 \coloneqq X \to X_1 \coloneqq\Sigma X \to X_2 \coloneqq\Sigma^2 X \to \cdots},\end{align*} yields a functor \(\Sigma^\infty:\text{Top} \to \text{PreSpectra}\) with adjoint \(\Omega^\infty\) where \(X \mapsto X_0\).
    • \(HG \coloneqq K(G, 0) \xrightarrow{\omega_0^*} K(G, 1) \to \cdots\) where we take the adjoint of the homotopy equivalences \(\omega_n: K(G, n) \to \OmegaK(G, n+1)\).
    • \(K{\hbox{-}}\)theory, where \(K = \qty{{\mathbf{Z}}\times BU \to U \to {\mathbf{Z}}\times BU \to U \to \cdots}\) using the equivalences given by 2-fold Bott periodicity.
    • Real \(K{\hbox{-}}\)theory \(KO = \qty{{\mathbf{Z}}\times BO \to \cdots}\) using 8-fold Bott periodicity.
    • Any infinite loop space \(X= X_0\), i.e. where \(X \simeq\Omega^k X_k\), then take \(X_0 \to X_1 \to \cdots\).
    • Function spectrum \(F(X, Y) = ?\).

  • Coproduct in \(\text{Top}_*\) is wedge

  • \(\text{Top}_*\) is a closed symmetric monoidal category, where the symmetric monoidal product is the smash product \(A \smash B\) for which there is a homeomorphism \(\hom_\top(A\smash B, C) \cong \hom_\top(A, \hom_\top(B, C))\).

  • Alternate definition of hoTop/DTop as localizing equivalence: initial category receiving a functor which sends blah equivalences to isomorphisms

  • The “usual” category of spectra ios the homotopy category of spectra. Triangulated with shift functor \(\Sigma({-})\) with inverse \(\Omega({-})\).

  • The sphere spectrum \(S^0\) is the unit for the symmetric monoidal structure, i.e. \(S^0 \smash X \simeq X, F(S^0, X) \simeq X\).

  • Pushout and pullback diagrams coincide, exact triangles \(X\to Y \to Z \to \Sigma X\) are equivalently fiber and cofiber sequences.

    • \(X\to Y\) is null-homotopic iff \(Z \simeq Y \vee \Sigma X\).

  • Spectra as generalized homology theories: take coproducts to direct sums and exact triangles to exact sequences

    • For \(E\) a spectrum, the homology theory is \(E^n(X) = \pi_n(E\smash X)\).
    • For \(E = HG\), \(HG^{-}(A) = \tilde H^{-}(X; G) \cong HG^{-}(\Sigma^\infty X)\).

  • Homotopy groups are well-defined for any spectrum, can be nonzero in negative degrees

  • Connective spectra: related to stages of Whitehead tower

  • Ring spectra: cohomology theories have a multiplicative structure, gives rise to maps \(H{\mathbf{Z}}\smash H{\mathbf{Z}}\to H{\mathbf{Z}}\). For any spectrum \(R\) with

    • A multiplication map \(R\smash R \to R\)
    • A unit map \(S^0 \to R\)
    • Require that this diagram commutes:
      \begin{center}
\begin{tikzcd}
S^0 \smash R \ar[r]\ar[dr, "\simeq"] & R\smash R\ar[d] & \ar[l] R\smash S\ar[dl, "\simeq"] \\
& R &
\end{tikzcd}
\end{center}
    • Commutativity: require that the swap map commutes with multiplication

  • Thom spectra: let \(\nu: E\to B\) be a real vector bundle over a space, take 1-point compactification of fibers to get the sphere bundle \(\text{Sph}(E) \to B\), take the section \(s(b) = \infty\) in each fiber, and define the Thom space as \(B^\nu = \text{Sph}(E)/s(B)\).

    • Take the suspension spectrum to get the Thom spectrum.

  • Virtual bundle: a formal difference of two bundles over a common base \(B\)

  • Model for \(BO_n = \colim_{\to k} {\operatorname{Gr}}(n, {\mathbf{R}}^k)\)

    • Can take the universal bundle \begin{align*}E_n = \left\{{(V, \mathbf{x}) \in G_n \times{\mathbf{R}}^\infty {~\mathrel{\Big\vert}~}\mathbf{x} \in V \in G_n }\right\} \to BO_n\end{align*}
    • \(MO_n\) is the associated Thom space

  • Cohomology operation of degree \(k\): a natural transformation \(H^*({-}; {\mathbf{Z}}/2{\mathbf{Z}}) \to H^{*+k}({-}; {\mathbf{Z}}/2{\mathbf{Z}})\). Stable if if commutes with the suspension isomorphism \(H^*({-}) \cong H^{*+1}(\Sigma({-}))\).

    • Example: Bockstein morphism, take \(0\to A \to B \to C \to 0\) in abelian groups to get a LES, the connecting morphism \(H^*({-}; {\mathbf{Z}}/2{\mathbf{Z}}) \to H^{*+1}({-}, {\mathbf{Z}}/2{\mathbf{Z}})\) is \(\text{Sq}^1\), a stable operation of degree 1.

  • Can form \({\mathbf{RP}}^2\) as a pushout: 

    \begin{center}
\begin{tikzcd}
S^1 \arrow[dd] \arrow[rr] &  & D^2 \arrow[dd, dashed] \\
                          &  &                        \\
S^1 \arrow[rr, dashed]    &  & {\mathbf{RP}}^2                 
\end{tikzcd}
\end{center}

    and \({\mathbf{CP}}^2\) as 

    \begin{center}
\begin{tikzcd}
S^3 \arrow[dd, "\eta"] \arrow[rr] &  & D^4 \arrow[dd, dashed] \\
                          &  &                        \\
S^2 \arrow[rr, dashed]    &  & {\mathbf{CP}}^2                 
\end{tikzcd}
\end{center}

    where \(\eta\) is the Hopf fibration.

  • Steenrod algebra \({\mathcal{A}}\): graded, non-commutative (cocommutative Hopf) \({ \mathbf{F} }_2{\hbox{-}}\)algebra generated in degree \(k\) by stable cohomology operations of degree \(k\), multiplication given by composition of operations

    • Use Yoneda to show \({\mathcal{A}}\cong H{\mathbf{Z}}/2{\mathbf{Z}}^*(H{\mathbf{Z}}/2{\mathbf{Z}})\).
    • Axiomatically describe squares as \(\text{Sq}^k: H^*({-}; {\mathbf{Z}}/2{\mathbf{Z}}) \to H^{*+k}({-}; {\mathbf{Z}}/2{\mathbf{Z}})\).

  • Adams Spectral Sequence: in good cases, \begin{align*}E_2^{s, t} = \operatorname{Ext} _{{\mathcal{A}}}^{s, t}(H^*(X), {\mathbf{Z}}/2{\mathbf{Z}}) \implies \qty{\pi_{t-s}X}^{\smash}_2\end{align*}

  • Hurewicz morphism: send a map \(f: S^k \to X\) to its induced map on cohomology.

  • Generalized EM spectrum: \(Z \simeq HV \simeq\bigvee_{i\in I} \Sigma^i H{\mathbf{Z}}/2{\mathbf{Z}}\) where \(V\) is a graded \({\mathbf{Z}}/2{\mathbf{Z}}{\hbox{-}}\)vector space which is finite in every degree.

  • Idea: for \(X\) a suspension spectrum of a CW complex with finitely many cells in each dimension, resolve it (Adams resolution) by a sequence of spectra \(X_n\) admitting maps to generalized EM spectra.

    • For such spectra, the Adams SS computes the 2-completion of the homotopy groups of \(X\)
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