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 $k$th 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, proved that the $k$th group homology $$H_k(\Sigma_n)$$ of the symmetric group $$\Sigma_n$$ on $n$ elements is independent of $n$, as long as $$n \ge 2k$$. This is an instance of homological stability.

The Barratt–Priddy theorem states that these "stable homology groups" are the same: for $$n \ge 2k$$, 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 $$H_1(\Sigma_n) \cong \Z/2\Z$$, the cyclic group of order 2, for all $$n\ge 2$$. (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:
 * $$\pi_1(\operatorname{Map}_0(S^2,S^2))=H_1(\operatorname{Map}_0(S^2,S^2))\cong \Z$$.

A generator for this group can be built from the Hopf fibration $$S^3 \to S^2$$. Finally, once $$n\ge 3$$, both are cyclic of order 2:
 * $$\pi_1(\operatorname{Map}_0(S^n,S^n))=H_1(\operatorname{Map}_0(S^n,S^n))\cong \Z/2\Z$$.

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 $$n\ge 2k$$,
 * $$H_k(\Sigma_{\infty}) \cong H_k(\Sigma_{n})$$.

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 $$\R^{\infty}$$ 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 $Map_{0}(S^{∞},S^{∞})$. (Since $&pi;_{1}(Map_{0}(S^{∞},S^{∞}))≅H_{1}(Σ_{∞})≅Z/2Z$, the map $φ: BΣ_{∞}→Map_{0}(S^{∞},S^{∞})$ satisfies the universal property of the plus construction once it is known that $φ$ is a homology equivalence.)

The mapping spaces $Map_{0}(S^{n},S^{n})$ are more commonly denoted by $Ω^{n}_{0}S^{n}$, where $Ω^{n}S^{n}$ is the $n$-fold loop space of the $n$-sphere $S^{n}$, and similarly $Map_{0}(S^{∞},S^{∞})$ is denoted by $Ω^{∞}_{0}S^{∞}$. 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 $GL_{n}(F_{1})$ 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 $K_{i}(F_{1})$ of $F_{1}$ should be defined as $&pi;_{i}(BGL_{∞}(F_{1})^{+})=&pi;_{i}(BΣ_{∞}^{+})$, which by the Barratt–Priddy theorem is:
 * $$K_i(\mathbf{F}_1)=\pi_i(BGL_\infty(\mathbf{F}_1)^+)=\pi_i(B\Sigma_\infty^+)=\pi_i^s.$$