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Snaith theorem

Contents

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

The original theorem by Snaith (Snaith 79) identifies the complex K-theory spectrum KU simply as the localization of the ∞-group ∞-ring 𝕊[BU(1)]\mathbb{S}[B U(1)] of the circle 2-group away from the Bott element β\beta:

KU𝕊[BU(1)][β 1]. KU \simeq \mathbb{S}[B U(1)][\beta^{-1}] \,.

Later more instances of such characterizations of familiar E-∞ rings have been given:

Statement

Preliminaries

For AA an abelian ∞-group write E𝕊[A]=Σ + AE \coloneqq \mathbb{S}[A] = \Sigma^\infty_+ A for its ∞-group E-∞ ring.

Definition

For βπ n(E)\beta \in \pi_n(E) an element of the nnth stable homotopy group, then multiplication by β\beta is a homomorphism

β *:EΣ nE. \beta_\ast \;\colon\; E \to \Sigma^{-n} E \,.

The localization of EE at β\beta is the homotopy colimit over the iterated multiplication with β\beta

E[β 1]lim[Eβ *Σ nEΣ nβ *Σ 2nE] E[\beta^{-1}] \coloneqq \underset{\to}{\lim} \left[ E \stackrel{\beta_\ast}{\to} \Sigma^{-n}E \stackrel{\Sigma^{-n} \beta_\ast}{\to} \Sigma^{-2n} E \to \cdots \right]

which has the universal property that μ β\mu_\beta becomes an equivalence on E[β 1]E[\beta^{-1}].

For complex topological K-theory

The original formulation of Snaith’s theorem (Snaith 79, theorem 2.12, spring) for complex topological K-theory.

Preliminaries

Write (Ho(Top),×,*)(Ho(Top), \times, \ast) for the classical homotopy category, regarded as a symmetric monoidal category under forming product spaces, with tensor unit the point space.

Write (Ho(Top */),,S 0)(Ho(Top^{\ast/}), \wedge , S^0) for the homotopy category of pointed topological spaces with tensor product the smash product of pointed spaces and tensor unit the 0-sphere

Write (Ho(Spectra),,𝕊)(Ho(Spectra), \wedge, \mathbb{S}) be the stable homotopy category with its symmetric monoidal smash product of spectra \wedge whose tensor unit is the sphere spectrum 𝕊\mathbb{S}.

For A,BHo(Spectra)A,B \in Ho(Spectra) two spectra, we write

[A,B]Hom Ho(Spectra)(A,B)Ab [A,B] \coloneqq Hom_{Ho(Spectra)}(A,B) \in Ab

for hom-set in the stable homotopy category and write

[A,B] [Σ A,B] [A,B]_\bullet \coloneqq [\Sigma^\bullet A, B]

for the corresponding \mathbb{Z}-graded group (this def.).

There are two pairs of (derived) adjoint functors

Ho(Spectra)Ω Σ Ho(Top */)U() +Ho(Top) Ho(Spectra) \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty}{\longleftarrow}} {} Ho(Top^{\ast/}) \underoverset {\underset{U}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} Ho(Top)

which are strong monoidal functors (by this example and this prop). The left adjoint composite

𝕊[]:Σ (() +) \mathbb{S}[-] \;\colon \Sigma^\infty((-)_+)

hence takes H-spaces and in particular H-groups GG to ring spectra.

More in detail, an H-space structure is a GHo(Top)G \in Ho(Top) equipped with morphisms

μ:G×GG\mu \;\colon\; G \times G \longrightarrow G

e:*Ge \;\colon\; \ast \longrightarrow G

satisfying associativity and unitality in Ho(Spectra)Ho(Spectra), and the corresponding ring spectrum has product

(Σ (G +))(Σ G +)Σ ((G×G) +)Σ (e +)Σ (G +) \left(\Sigma^\infty(G_+)\right) \wedge \left( \Sigma^\infty G_+\right) \;\simeq\; \Sigma^\infty( (G \times G)_+ ) \overset{\Sigma^\infty (e_+)}{\longrightarrow} \Sigma^\infty (G_+)

and unit

𝕊Σ (* +)Σ (e +)Σ (G +) \mathbb{S} \simeq \Sigma^\infty( \ast_+) \overset{ \Sigma^\infty(e_+) }{\longrightarrow} \Sigma^\infty(G_+)

We call

𝕊[G]Σ (G +) \mathbb{S}[G] \coloneqq \Sigma^\infty(G_+)

equipped with this monoid structure the H-group ring spectrum of GG. See there for more.

The ring spectrum of the circle 2-group

One such H-group is the circle 2-group, hence (the homotopy type of) the classifying space BU(1)B U(1) for complex line bundles, equivalently the Eilenberg-MacLane space K(,2)K(\mathbb{Z},2), canonically presented by the complex projective space P \mathbb{C}P^\infty

K(,2)BU(1)P Ho(Spaces). K(\mathbb{Z},2) \simeq B U(1) \simeq \mathbb{C}P^\infty \;\in\; Ho(Spaces) \,.

This being the classifying space for complex line bundles, it becomes an H-group via the map

BU(1)×BU(1)BU(1) B U(1) \times B U(1) \longrightarrow B U(1)

which classifies the tensor product of line bundles, with inverses given by the map

BU(1)BU(1) B U(1) \longrightarrow B U(1)

which form dual line bundles.

Hence its H-group ring spectrum is

𝕊[BU(1)]=Σ (BU(1) +). \mathbb{S}[B U(1)] \;=\; \Sigma^\infty( B U(1)_+ ) \,.

Therefore for XHo(Top */)X \in Ho(Top^{\ast/}) a pointed topological space, then

[Σ X,Σ (G +)] [\Sigma^\infty X,\Sigma^\infty(G_+)]_\bullet

is a graded ring, with the product of elements

(Σ n iΣ Xα iΣ (G +))[Σ X,Σ (G +)] n i \left( \Sigma^{n_i} \Sigma^\infty X \overset{\alpha_i}{\longrightarrow} \Sigma^\infty(G_+) \right) \;\in\; [\Sigma^\infty X, \Sigma^\infty(G_+)]_{n_i}

for i{1,2}i \in \{1,2\} given by

Σ n 1+n 2Σ XΣ n 1+n 2Σ Δ XΣ n 1+n 2Σ (XX)(Σ n 1Σ X)(Σ n 2Σ X)α 1α 2(Σ (G +))(Σ (G +))Σ (μ +)Σ (G +). \Sigma^{n_1 + n_2} \Sigma^\infty X \overset{\Sigma^{n_1+ n_2} \Sigma^\infty \Delta_X}{\longrightarrow} \Sigma^{n_1 + n_2} \Sigma^\infty (X \wedge X) \simeq \left( \Sigma^{n_1} \Sigma^\infty X \right) \wedge \left( \Sigma^{n_2} \Sigma^\infty X \right) \overset{ \alpha_1 \wedge \alpha_2 }{\longrightarrow} \left( \Sigma^\infty (G_+) \right) \wedge \left( \Sigma^\infty (G_+) \right) \overset{\Sigma^\infty (\mu_+)}{\longrightarrow} \Sigma^\infty (G_+) \,.

Here the isomorphism on the left is the combination of the strong monoidalness of Σ \Sigma^\infty with the respect of suspension Σ\Sigma for the smash product of spectra (the tensor triangulated category structure on Ho(Spectra)Ho(Spectra), this prop.).

Observe that we have a splitting

Σ (BU(1) +)(Σ (BU(1)))𝕊 \Sigma^\infty(B U(1)_+) \;\simeq\; \left( \Sigma^\infty (B U(1)) \right) \;\oplus\; \mathbb{S}

(by this remark) and hence a canonical morphism

Σ (BU(1))Σ (BU(1) +). \Sigma^\infty (B U(1)) \longrightarrow \Sigma^\infty (B U(1)_+) \,.

Via this splitting, the morphism in Ho(Top */)Ho(Top^{\ast/})

h:S 2BU(1) h \colon S^2 \longrightarrow B U(1)

in [S 2,BU(1)]π 2(BU(1))[S^2, B U(1)] \simeq \pi_2(B U(1)) \simeq \mathbb{Z} which classifies the basic complex line bundle on the 2-sphere and represents 11 \in \mathbb{Z}, induces a morphism

β:Σ 2𝕊Σ S 2Σ (h)Σ BU(1)Σ (BU(1) +) \beta \;\colon\; \Sigma^2 \mathbb{S} \simeq \Sigma^\infty S^2 \overset{ \Sigma^\infty( h ) }{\longrightarrow} \Sigma^\infty B U(1) \overset{}{\longrightarrow} \Sigma^\infty (B U(1)_+)

hence an element in [𝕊,Σ (BU(1) +)] 2[\mathbb{S}, \Sigma^\infty(B U(1)_+)]_2.

The map to the K-theory spectrum

The spectrum KU which represents complex topological K-theory has in degree 0 the the product space

Ω KUBU×Ho(Top */) \Omega^\infty KU \simeq B U \times \mathbb{Z} \in Ho(Top^{\ast/})

of the stable classifying space BUB U for complex vector bundles and the integers. The base point is (*,0)(\ast, 0).

The projection

BU× B U \times \mathbb{Z} \longrightarrow \mathbb{Z}

classifies the virtual rank of virtual vector bundle.

Hence the inclusion of classifying spaces

BUBU×{0}BU× B U \simeq B U \times \{0\} \hookrightarrow B U \times \mathbb{Z}

classifies the inclusion K˜ ()K ()\tilde K_{\mathbb{C}}(-) \hookrightarrow K_{\mathbb{C}}(-) of reduced K-theory.

There is a canonical morphism

Σ (BU(1) +)ΦKU \Sigma^\infty(B U(1)_+) \overset{\Phi}{\longrightarrow} K U

in Ho(Spectra)Ho(Spectra), being the (Σ Ω )(\Sigma^\infty \dashv \Omega^\infty)-adjunct of

BU(1) +BU×Ω KU B U(1)_+ \longrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U

in Ho(Top */)Ho(Top^{\ast/}), which in turn is the (() +U)((-)_+ \dashv U)-adjunct of the canonical

𝒪(1):BU(1)BUBU×{1}BU× \mathcal{O}(1) \;\colon\; B U(1) \longrightarrow B U \simeq B U \times \{1\} \hookrightarrow B U \times \mathbb{Z}

in Ho(Top)Ho(Top).

Since the formal group law for K-theory says that μ *(𝒪(1))pr 1 *(𝒪(1))pr 2 *(𝒪(2))\mu^\ast ( \mathcal{O}(1) ) \simeq pr_1^\ast (\mathcal{O}(1)) \otimes pr_2^\ast(\mathcal{O}(2)) Φ\Phi is a homomoprhism of ring spectra.

Under the above splitting, the morphism Φ\Phi decomposes as

Φ:Σ (BU(1) +)Σ BU(1)𝕊(i˜,e KU)KU \Phi \;\colon\; \Sigma^\infty (B U(1)_+) \simeq \Sigma^\infty B U(1) \oplus \mathbb{S} \overset{ ( \tilde i , e_{KU}) }{\longrightarrow} KU

Since, by the above, morphisms Σ BU(1)KU\Sigma^\infty B U(1) \longrightarrow K U in Ho(Spectra)Ho(Spectra), hence equivalently morphisms BU(1)BU×B U(1) \longrightarrow B U \times \mathbb{Z} in Ho(Top */)Ho(Top^{\ast/}), hence equivalently morphisms BU(1)BUB U(1) \to B U in Ho(Top)Ho(Top) correspond to the reduced K-theory of BU(1)B U(1), and since morphisms 𝕊KU\mathbb{S} \to KU in Ho(Spectra)Ho(Spectra), hence equivalently morphism *BU×\ast \to B U \times \mathbb{Z} in Ho(Top)Ho(Top) correspond to the K-theory of the point, and since over (colimits of) compact topological spaces K-theory splits as K(X)K˜(X)K(*)K(X) \simeq \tilde K(X) \oplus K(\ast) via [E]n([E]rk(E))+(rk(E)n)[E]- n \mapsto ([E] - rk(E)) + ( rk(E) - n ) (this prop) it follows that

  1. i˜\tilde i takes the canonical line bundle 𝒪(1)\mathcal{O}(1) on BU(1)B U(1) to its image 𝒪(1)1\mathcal{O}(1)-1 in reduced K-theory

    BU(1)𝒪(1)(𝒪(1)1)BUBU×{0}BU×Ω KU B U(1) \overset{\mathcal{O}(1) \mapsto (\mathcal{O}(1)-1)}{\longrightarrow} B U \simeq B U \times \{0\} \hookrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U
  2. e KUe_{K U} is adjunct to *(*,1)BU×\ast \simeq (\ast,1) \hookrightarrow B U \times \mathbb{Z} ( is the ring spectrum unit of KUK U).

Now observe that Φ\Phi takes multiplication by β\beta to multiplication with the Bott element (h1)(h-1):

This is because multiplication by β\beta is the outer right boundary of the following diagram

Σ S 2Σ nΣ X (h0)id (Σ (BU(1))𝕊)Σ nΣ X idα Σ (BU(1) +)Σ nΣ (BU(1) +) Σ nΣ (μ +) Σ nΣ (BU(1) +) KUΣ nKU Σ nKU \array{ \Sigma^\infty S^2 \wedge \Sigma^n \Sigma^\infty X \\ {}^{\mathllap{ (h \oplus 0) \wedge id }}\downarrow \\ \left(\Sigma^\infty( B U(1) ) \oplus \mathbb{S}\right) \wedge \Sigma^n \Sigma^\infty X \\ {}^{\mathllap{id \wedge \alpha}}\downarrow \\ \Sigma^\infty(B U(1)_+) \wedge \Sigma^n \Sigma^\infty( B U(1)_+ ) &\overset{\Sigma^n\Sigma^\infty (\mu_+)}{\longrightarrow}& \Sigma^n \Sigma^\infty (B U(1)_+) \\ \downarrow && \downarrow \\ K U \wedge \Sigma^n K U &\longrightarrow & \Sigma^n K U }

and since the bottom square commutes (since tensor product of line bundles corresponds to their product in K-theory) this is equivalent to the left and bottom boundary, which, by the above discussion, is multiplication with the Bott element (h1)(h-1).

Since the Bott element is invertible in KUK U, this means for all XHo(Top */)X \in Ho(Top^\ast/) that the morphism

[Σ X,Σ (B(1) +)] Φ XΦ()[Σ X,KU] K˜ (X) [\Sigma^\infty X , \Sigma^\infty ( B (1)_+ ) ]_\bullet \overset{\Phi_X \coloneqq \Phi \circ (-)}{\longrightarrow} [\Sigma^\infty X, K U]_\bullet \simeq \tilde K_{\mathbb{C}}^\bullet(X)

extends to the quotient ring

[Σ X,Σ (BU(1) +)] [β 1] [\Sigma^\infty X, \Sigma^\infty ( B U(1)_+ )]_{\bullet}[\beta^{-1}]

in which two elements are identified if they differ by multiplication by β\beta, as above:

[Σ X,Σ (BU(1) +)] Φ K˜ (X) Φ [Σ X,Σ (BU(1) +)] [β 1]. \array{ [\Sigma^\infty X, \Sigma^\infty( B U(1)_+ )] &\overset{\Phi}{\longrightarrow}& \tilde K_{\mathbb{C}}^\bullet(X) \\ \downarrow & \nearrow_{\mathrlap{\exists \Phi}} \\ [\Sigma^\infty X, \Sigma^\infty( B U(1)_+ )]_\bullet [\beta^{-1}] } \,.

Isomorphy

Theorem

If XHo(Top */)X \in Ho(Top^{\ast/}) has the homotopy type of a finite CW-complex, then the natural transformation

Φ X:[X,Σ (BU(1) +)] [β 1]K˜ (X) \Phi_X \;\colon\; [X, \Sigma^\infty ( B U(1)_+ )]_\bullet [\beta^{-1}] \longrightarrow \tilde K^\bullet(X)

is a natural isomorphism.

(Snaith 81, theorem 2.12, Hopkins-Mathew)

That (before localization) the map is an epimorphism is due to (Segal 73, prop. 1), see this prop.. The analog of this statement for real projective space P B/2\mathbb{R}P^\infty \simeq B \mathbb{Z}/2 instead of complex projective space P BU(1)\mathbb{C}P^\infty \simeq B U(1) is the Kahn-Priddy theorem.

For periodic complex cobordism

Theorem

The periodic complex cobordism spectrum is the ∞-group ∞-ring of the classifying space for stable complex vector bundles (the classifying space for topological K-theory) localized at the Bott element β\beta:

PMU(𝕊[BU])[β 1]. PMU \simeq (\mathbb{S}[B U])[\beta^{-1}] \,.

(Snaith 81, theorem 2.7)

For algebraic K-theory

Remark

The analog of this result for the periodic algebraic cobordism spectrum? and algebraic K-theory as motivic spectra is discussed in (GepnerSnaith 08).

For smooth spectra and differential K-theory

Refinement of the Snaith theorem for KU to smooth spectra and to differential K-theory is in (Bunke-Nikolaus-Völkl 13, section 6.3). See at differential cohomology diagram – Smooth Snaith K-theory.

Snaith-like theorem for Morava EE-theories

Write Γ n\Gamma_n for the Honda formal group. The automorphism group Aut(Γ n)Aut(\Gamma_n) induces for each prime pp a canonical determinant morphism

det:𝔾 n p × det \;\colon\; \mathbb{G}_n \longrightarrow \mathbb{Z}_p^\times

from the Morava stabilizer group 𝔾 nGal(𝔽 p n/𝔽 p)Aut(Γ n)\mathbb{G}_n \coloneqq Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) \ltimes Aut(\Gamma_n).

Write

S𝔾 nker(det) S \mathbb{G}_n \coloneqq ker(det)

for the kernel. This naturally acts on the Morava E-theory spectrum E nE_n. Write E S𝔾 nE^{S\mathbb{G}_n} for the corresponding homotopy fixed point spectrum. (Westerland 12, 1.1).

Write L K(n)𝕊[B n+1 p]L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p] for the K(n)K(n)-localization of the ∞-group ∞-ring of the (n+1)-group B n pB^n \mathbb{Z}_p.

Theorem

There is a generalized element ρ n\rho_n of the E-∞ ring L K(n)𝕊[B n+1 p]L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p] such that localization at that element yields the Morava E-theory spectrum S𝔾 nS\mathbb{G}_nhomotopy fixed points:

L K(n)𝕊[B n+1 p][ρ n 1]E n S𝔾 n. L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p] [\rho_n^{-1}] \stackrel{\simeq}{\longrightarrow} E_n^{S\mathbb{G}_n} \,.

(Westerland 12, theorem 1.2)

References

The theorem is due to

  • Victor Snaith, Algebraic Cobordism and K-theory, Mem. Amer. Math. Soc. no 221 (1979)

with a simpler proof given in

  • Victor Snaith, Localized stable homotopy of some classifying spaces, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2,

    325-330. MR 600247 (82g:55006) (pdf)

using results from

Another proof due to Mike Hopkins is in

Refinement to smooth spectra and differential K-theory is in

Discussion of the E-∞ ring structure involved is around theorem 3.1 of

A version for motivic spectra algebraic K-theory is discussed in

and for motivic cohomology in

Higher chromatic analogs for Morava E-theory are discussed in

See also at spherical T-duality.

Last revised on February 22, 2018 at 15:32:44. See the history of this page for a list of all contributions to it.