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 ${\displaystyle \operatorname {Map} _{0}(S^{n},S^{n})}$ is the topological space of all continuous maps ${\displaystyle f\colon S^{n}\to S^{n}}$ from the n-dimensional sphere ${\displaystyle S^{n}}$ to itself, under the topology of uniform convergence (a special case of the compact-open topology). These maps are required to fix a basepoint ${\displaystyle x\in S^{n}}$, satisfying ${\displaystyle f(x)=x}$, 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 ${\displaystyle \Sigma _{n}}$.

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

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

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

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

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

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

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

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

Reformulation of the theorem

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

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

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

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

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

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

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

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

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

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

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

"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

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

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:

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

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
• Nakaoka, Minoru (1960), "Decomposition theorem for homology groups of symmetric groups", Annals of Mathematics, 71: 16–42, doi:10.2307/1969878, JSTOR 1969878, MR 0112134