Barratt–Priddy theorem

In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem (also referred to as Barratt–Priddy–Quillen theorem) expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem (named after Michael Barratt, Stewart Priddy, and Daniel Quillen) is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction.

Statement of the theorem

The mapping space $\operatorname {Map} _{0}(S^{n},S^{n})$ is the topological space of all continuous maps $f\colon S^{n}\to S^{n}$ from the n-dimensional sphere $S^{n}$ to itself, under the topology of uniform convergence (a special case of the compact-open topology). These maps are required to fix a basepoint $x\in S^{n}$, satisfying $f(x)=x$, and to have degree 0; this guarantees that the mapping space is connected. The Barratt–Priddy theorem expresses a relation between the homology of these mapping spaces and the homology of the symmetric groups $\Sigma _{n}$.

It follows from the Freudenthal suspension theorem and the Hurewicz theorem that the kth homology $H_{k}(\operatorname {Map} _{0}(S^{n},S^{n}))$ of this mapping space is independent of the dimension n, as long as $n>k$. Similarly, Minoru Nakaoka (1960) proved that the kth group homology $H_{k}(\Sigma _{n})$ of the symmetric group $\Sigma _{n}$ on n elements is independent of n, as long as $n\geq 2k$. This is an instance of homological stability.

The Barratt–Priddy theorem states that these "stable homology groups" are the same: for $n\geq 2k$, there is a natural isomorphism

$H_{k}(\Sigma _{n})\cong H_{k}({\text{Map}}_{0}(S^{n},S^{n})).$

This isomorphism holds with integral coefficients (in fact with any coefficients, as is made clear in the reformulation below).

Example: first homology

This isomorphism can be seen explicitly for the first homology $H_{1}$. The first homology of a group is the largest commutative quotient of that group. For the permutation groups $\Sigma _{n}$, the only commutative quotient is given by the sign of a permutation, taking values in {−1, 1}. This shows that $H_{1}(\Sigma _{n})\cong \mathbb {Z} /2\mathbb {Z}$, the cyclic group of order 2, for all $n\geq 2$. (For $n=1$, $\Sigma _{1}$ is the trivial group, so $H_{1}(\Sigma _{1})=0$.)

It follows from the theory of covering spaces that the mapping space $\operatorname {Map} _{0}(S^{1},S^{1})$ of the circle $S^{1}$ is contractible, so $H_{1}(\operatorname {Map} _{0}(S^{1},S^{1}))=0$. For the 2-sphere $S^{2}$, the first homotopy group and first homology group of the mapping space are both infinite cyclic:

$\pi _{1}(\operatorname {Map} _{0}(S^{2},S^{2}))=H_{1}(\operatorname {Map} _{0}(S^{2},S^{2}))\cong \mathbb {Z}$.

A generator for this group can be built from the Hopf fibration $S^{3}\to S^{2}$. Finally, once $n\geq 3$, both are cyclic of order 2:

$\pi _{1}(\operatorname {Map} _{0}(S^{n},S^{n}))=H_{1}(\operatorname {Map} _{0}(S^{n},S^{n}))\cong \mathbb {Z} /2\mathbb {Z}$.

Reformulation of the theorem

The infinite symmetric group $\Sigma _{\infty }$ is the union of the finite symmetric groups $\Sigma _{n}$, and Nakaoka's theorem implies that the group homology of $\Sigma _{\infty }$ is the stable homology of $\Sigma _{n}$: for $n\geq 2k$,

$H_{k}(\Sigma _{\infty })\cong H_{k}(\Sigma _{n})$.

The classifying space of this group is denoted $B\Sigma _{\infty }$, and its homology of this space is the group homology of $\Sigma _{\infty }$:

$H_{k}(B\Sigma _{\infty })\cong H_{k}(\Sigma _{\infty })$.

We similarly denote by $\operatorname {Map} _{0}(S^{\infty },S^{\infty })$ the union of the mapping spaces $\operatorname {Map} _{0}(S^{n},S^{n})$ under the inclusions induced by suspension. The homology of $\operatorname {Map} _{0}(S^{\infty },S^{\infty })$ is the stable homology of the previous mapping spaces: for $n>k$,

$H_{k}(\operatorname {Map} _{0}(S^{\infty },S^{\infty }))\cong H_{k}(\operatorname {Map} _{0}(S^{n},S^{n})).$

There is a natural map $\varphi \colon B\Sigma _{\infty }\to \operatorname {Map} _{0}(S^{\infty },S^{\infty })$; one way to construct this map is via the model of $B\Sigma _{\infty }$ as the space of finite subsets of $\mathbb {R} ^{\infty }$ endowed with an appropriate topology. An equivalent formulation of the Barratt–Priddy theorem is that $\varphi$ is a homology equivalence (or acyclic map), meaning that $\varphi$ induces an isomorphism on all homology groups with any local coefficient system.

Relation with Quillen's plus construction

The Barratt–Priddy theorem implies that the space BΣ_∞⁺ resulting from applying Quillen's plus construction to BΣ_∞ can be identified with Map₀(S^∞,S^∞). (Since π₁(Map₀(S^∞,S^∞))≅H₁(Σ_∞)≅Z/2Z, the map φ: BΣ_∞→Map₀(S^∞,S^∞) satisfies the universal property of the plus construction once it is known that φ is a homology equivalence.)

The mapping spaces Map₀(Sⁿ,Sⁿ) are more commonly denoted by Ωⁿ₀Sⁿ, where ΩⁿSⁿ is the n-fold loop space of the n-sphere Sⁿ, and similarly Map₀(S^∞,S^∞) is denoted by Ω^∞₀S^∞. Therefore the Barratt–Priddy theorem can also be stated as

$B\Sigma _{\infty }^{+}\simeq \Omega _{0}^{\infty }S^{\infty }$ or

${\textbf {Z}}\times B\Sigma _{\infty }^{+}\simeq \Omega ^{\infty }S^{\infty }$

In particular, the homotopy groups of BΣ_∞⁺ are the stable homotopy groups of spheres:

$\pi _{i}(B\Sigma _{\infty }^{+})\cong \pi _{i}(\Omega ^{\infty }S^{\infty })\cong \lim _{n\rightarrow \infty }\pi _{n+i}(S^{n})=\pi _{i}^{s}(S^{n})$

"K-theory of F₁"

The Barratt–Priddy theorem is sometimes colloquially rephrased as saying that "the K-groups of F₁ are the stable homotopy groups of spheres". This is not a meaningful mathematical statement, but a metaphor expressing an analogy with algebraic K-theory.

The "field with one element" F₁ is not a mathematical object; it refers to a collection of analogies between algebra and combinatorics. One central analogy is the idea that GL_n(F₁) should be the symmetric group Σ_n. The higher K-groups K_i(R) of a ring R can be defined as

$K_{i}(R)=\pi _{i}(BGL_{\infty }(R)^{+})$

According to this analogy, the K-groups K_i(F₁) of F₁ should be defined as π_i(BGL_∞(F₁)⁺)=π_i(BΣ_∞⁺), which by the Barratt–Priddy theorem is:

$K_{i}(\mathbf {F} _{1})=\pi _{i}(BGL_{\infty }(\mathbf {F} _{1})^{+})=\pi _{i}(B\Sigma _{\infty }^{+})=\pi _{i}^{s}.$

References

• Barratt, Michael; Priddy, Stewart (1972), "On the homology of non-connected monoids and their associated groups", Commentarii Mathematici Helvetici, 47: 1–14, doi:10.1007/bf02566785, S2CID 119714992
• Nakaoka, Minoru (1960), "Decomposition theorem for homology groups of symmetric groups", Annals of Mathematics, 71 (1): 16–42, doi:10.2307/1969878, JSTOR 1969878, MR 0112134