# 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$_∞$$^+$ resulting from applying Quillen's plus construction to BΣ$_∞$ can be identified with Map$_0$(S$^∞$,S$^∞$). (Since π$_1$(Map$_0$(S$^∞$,S$^∞$))≅H$_1$(Σ$_∞$)≅Z/2Z, the map φ: BΣ$_∞$→Map$_0$(S$^∞$,S$^∞$) satisfies the universal property of the plus construction once it is known that φ is a homology equivalence.)
The mapping spaces Map$_0$(S$^n$,S$^n$) are more commonly denoted by Ω$^n$$_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 F$_1$"

The Barratt–Priddy theorem is sometimes colloquially rephrased as saying that "the K-groups of F$_1$ 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" F$_1$ 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 π$_i$(BGL$_∞$(F$_1$)$^+$)=π$_i$(BΣ$_∞$$^+$), which by the Barratt–Priddy theorem is:
    $K_{i}(\mathbf {F} _{1})=\pi _{i}(BGL_{\infty }(\mathbf {F} _{1})^{+})=\pi _{i}(B\Sigma _{\infty }^{+})=\pi _{i}^{s}.$