nLab Snaith theorem

Idea

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

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

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

• in (Gepner-Snaith 08) the analogous statement for algebraic K-theory is formulated in terms of motivic spectra;

• in (Westerland 12) the tower of Morava E-theories (or some homotopy fixed point spectra of these) is shown to be localizations of the ∞-group ∞-rings of $B^{n+1} \mathbb{Z}_p$, for all $n \in \mathbb{N}$.

Statement

Preliminaries

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

Definition

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

$$\beta_\ast \;\colon\; E \to \Sigma^{-n} E \,.$$

The localization of $E$ at $\beta$ is the homotopy colimit over the iterated multiplication with $\beta$

$$E[\beta^{-1}] \coloneqq \underset{\to}{\lim} \left[ E \stackrel{\beta_\ast}{\to} \Sigma^{-n}E \stackrel{\Sigma^{-n} \beta_\ast}{\to} \Sigma^{-2n} E \to \cdots \right]$$

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

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

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

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

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

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

for hom-set in the stable homotopy category and write

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

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

There are two pairs of (derived) adjoint functors

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

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

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

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

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

$\mu \;\colon\; G \times G \longrightarrow G$

$e \;\colon\; \ast \longrightarrow G$

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

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

and unit

$$\mathbb{S} \simeq \Sigma^\infty( \ast_+) \overset{ \Sigma^\infty(e_+) }{\longrightarrow} \Sigma^\infty(G_+)$$

We call

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

equipped with this monoid structure the H-group ring spectrum of $G$. 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 $B U(1)$ for complex line bundles, equivalently the Eilenberg-MacLane space $K(\mathbb{Z},2)$, canonically presented by the complex projective space $\mathbb{C}P^\infty$

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

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

$$B U(1) \times B U(1) \longrightarrow B U(1)$$

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

$$B U(1) \longrightarrow B U(1)$$

which form dual line bundles.

Hence its H-group ring spectrum is

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

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

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

is a graded ring, with the product of elements

$$\left( \Sigma^{n_i} \Sigma^\infty X \overset{\alpha_i}{\longrightarrow} \Sigma^\infty(G_+) \right) \;\in\; [\Sigma^\infty X, \Sigma^\infty(G_+)]_{n_i}$$

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

$$\Sigma^{n_1 + n_2} \Sigma^\infty X \overset{\Sigma^{n_1+ n_2} \Sigma^\infty \Delta_X}{\longrightarrow} \Sigma^{n_1 + n_2} \Sigma^\infty (X \wedge X) \simeq \left( \Sigma^{n_1} \Sigma^\infty X \right) \wedge \left( \Sigma^{n_2} \Sigma^\infty X \right) \overset{ \alpha_1 \wedge \alpha_2 }{\longrightarrow} \left( \Sigma^\infty (G_+) \right) \wedge \left( \Sigma^\infty (G_+) \right) \overset{\Sigma^\infty (\mu_+)}{\longrightarrow} \Sigma^\infty (G_+) \,.$$

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

Observe that we have a splitting

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

(by this remark) and hence a canonical morphism

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

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

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

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

$$\beta \;\colon\; \Sigma^2 \mathbb{S} \simeq \Sigma^\infty S^2 \overset{ \Sigma^\infty( h ) }{\longrightarrow} \Sigma^\infty B U(1) \overset{}{\longrightarrow} \Sigma^\infty (B U(1)_+)$$

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

The map to the K-theory spectrum

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

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

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

The projection

$$B U \times \mathbb{Z} \longrightarrow \mathbb{Z}$$

classifies the virtual rank of virtual vector bundle.

Hence the inclusion of classifying spaces

$$B U \simeq B U \times \{0\} \hookrightarrow B U \times \mathbb{Z}$$

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

There is a canonical morphism

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

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

$$B U(1)_+ \longrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U$$

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

$$\mathcal{O}(1) \;\colon\; B U(1) \longrightarrow B U \simeq B U \times \{1\} \hookrightarrow B U \times \mathbb{Z}$$

in $Ho(Top)$.

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

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

$$\Phi \;\colon\; \Sigma^\infty (B U(1)_+) \simeq \Sigma^\infty B U(1) \oplus \mathbb{S} \overset{ ( \tilde i , e_{KU}) }{\longrightarrow} KU$$

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

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

   $$B U(1) \overset{\mathcal{O}(1) \mapsto (\mathcal{O}(1)-1)}{\longrightarrow} B U \simeq B U \times \{0\} \hookrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U$$

2. $e_{K U}$ is adjunct to $\ast \simeq (\ast,1) \hookrightarrow B U \times \mathbb{Z}$ ( is the ring spectrum unit of $K U$).

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

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

$$\array{ \Sigma^\infty S^2 \wedge \Sigma^n \Sigma^\infty X \\ {}^{\mathllap{ (h \oplus 0) \wedge id }}\downarrow \\ \left(\Sigma^\infty( B U(1) ) \oplus \mathbb{S}\right) \wedge \Sigma^n \Sigma^\infty X \\ {}^{\mathllap{id \wedge \alpha}}\downarrow \\ \Sigma^\infty(B U(1)_+) \wedge \Sigma^n \Sigma^\infty( B U(1)_+ ) &\overset{\Sigma^n\Sigma^\infty (\mu_+)}{\longrightarrow}& \Sigma^n \Sigma^\infty (B U(1)_+) \\ \downarrow && \downarrow \\ K U \wedge \Sigma^n K U &\longrightarrow & \Sigma^n K U }$$

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 $(h-1)$.

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

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

extends to the quotient ring

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

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

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

Isomorphy

Theorem

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

$$\Phi_X \;\colon\; [X, \Sigma^\infty ( B U(1)_+ )]_\bullet [\beta^{-1}] \longrightarrow \tilde K^\bullet(X)$$

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 $\mathbb{R}P^\infty \simeq B \mathbb{Z}/2$ instead of complex projective space $\mathbb{C}P^\infty \simeq B U(1)$ 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 $\beta$:

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

(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 $E$-theories

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

$$det \;\colon\; \mathbb{G}_n \longrightarrow \mathbb{Z}_p^\times$$

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

Write

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

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

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

Theorem

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

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

(Westerland 12, theorem 1.2)

Related concepts

• splitting principle

• algebraic cobordism

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, doi:10.1017/S0305004100058205)

using results from

• Graeme Segal, The stable homotopy of complex of projective space, The quarterly journal of mathematics (1973) 24 (1): 1-5. (pdf, doi:10.1093/qmath/24.1.1)

Another proof due to Mike Hopkins is in

• Akhil Mathew (following Mike Hopkins), Snaith’s construction of complex K-theory (pdf, pdf)

Refinement to smooth spectra and differential K-theory is in

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, section 6.3 of Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

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

• Jacob Lurie, A Survey of Elliptic Cohomology

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

• David Gepner, Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic K-theory, Documenta Math. 2008 (arXiv:0712.2817)

and for motivic cohomology in

• Markus Spitzweck, Paul Arne Østvær, The Bott inverted infinite projective space is homotopy algebraic K-theory (pdf)

Higher chromatic analogs for Morava E-theory are discussed in

• Craig Westerland, A higher chromatic analogue of the image of J (arXiv:1210.2472)

• John Lind, Hisham Sati, Craig Westerland, A higher categorical analogue of topological T-duality for sphere bundles (arXiv:1601.06285)

A unifying general abstract perspective is discussed in

• Peter Arndt, section 3.2 of Abstract motivic homotopy theory, thesis 2017 (web, pdf, pdf)

  exposition: lecture at Geometry in Modal HoTT, 2019 (recording I, recording II)

See also at spherical T-duality.