Barratt–Priddy theorem

Template:Short description 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 ${\mathrm{Map}}_0(S^n,S^n)$ is the topological space of all continuous maps $f:S^n\to S^n$ from the Template:Mvar-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 ${\mathrm{\Sigma }}_n$.

It follows from the Freudenthal suspension theorem and the Hurewicz theorem that the Template:Mvarth homology $H_k({\mathrm{Map}}_0(S^n,S^n))$ of this mapping space is independent of the dimension Template:Mvar, as long as $n>k$. Similarly, Template:Harvs proved that the Template:Mvarth group homology $H_k({\mathrm{\Sigma }}_n)$ of the symmetric group ${\mathrm{\Sigma }}_n$ on Template:Mvar elements is independent of Template:Mvar, 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

$H_k({\mathrm{\Sigma }}_n)\cong H_k({\text{Map}}_0(S^n,S^n)).$

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 ${\mathrm{\Sigma }}_n$, the only commutative quotient is given by the sign of a permutation, taking values in Template:Math}. This shows that $H_1({\mathrm{\Sigma }}_n)\cong \mathbb{Z}/2\mathbb{Z}$, the cyclic group of order 2, for all $n\ge 2$. (For $n=1$, ${\mathrm{\Sigma }}_1$ is the trivial group, so $H_1({\mathrm{\Sigma }}_1)=0$.)

It follows from the theory of covering spaces that the mapping space ${\mathrm{Map}}_0(S^1,S^1)$ of the circle $S^1$ is contractible, so $H_1({\mathrm{Map}}_0(S^1,S^1))=0$. For the 2-sphere $S^2$, the first homotopy group and first homology group of the mapping space are both infinite cyclic:

${\pi }_1({\mathrm{Map}}_0(S^2,S^2))=H_1({\mathrm{Map}}_0(S^2,S^2))\cong \mathbb{Z}$.

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

${\pi }_1({\mathrm{Map}}_0(S^n,S^n))=H_1({\mathrm{Map}}_0(S^n,S^n))\cong \mathbb{Z}/2\mathbb{Z}$.

Reformulation of the theorem

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

$H_k({\mathrm{\Sigma }}_{\mathrm{\infty }})\cong H_k({\mathrm{\Sigma }}_n)$.

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

$H_k(B{\mathrm{\Sigma }}_{\mathrm{\infty }})\cong H_k({\mathrm{\Sigma }}_{\mathrm{\infty }})$.

We similarly denote by ${\mathrm{Map}}_0(S^{\mathrm{\infty }},S^{\mathrm{\infty }})$ the union of the mapping spaces ${\mathrm{Map}}_0(S^n,S^n)$ under the inclusions induced by suspension. The homology of ${\mathrm{Map}}_0(S^{\mathrm{\infty }},S^{\mathrm{\infty }})$ is the stable homology of the previous mapping spaces: for $n>k$,

$H_k({\mathrm{Map}}_0(S^{\mathrm{\infty }},S^{\mathrm{\infty }}))\cong H_k({\mathrm{Map}}_0(S^n,S^n)).$

There is a natural map $\phi :B{\mathrm{\Sigma }}_{\mathrm{\infty }}\to {\mathrm{Map}}_0(S^{\mathrm{\infty }},S^{\mathrm{\infty }})$; one way to construct this map is via the model of $B{\mathrm{\Sigma }}_{\mathrm{\infty }}$ as the space of finite subsets of ${ℝ}^{\mathrm{\infty }}$ endowed with an appropriate topology. An equivalent formulation of the Barratt–Priddy theorem is that $\phi$ is a homology equivalence (or acyclic map), meaning that $\phi$ 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 Template:Math resulting from applying Quillen's plus construction to Template:Math can be identified with Template:Math. (Since Template:Math, the map Template:Math satisfies the universal property of the plus construction once it is known that Template:Mvar is a homology equivalence.)

The mapping spaces Template:Math are more commonly denoted by Template:Math, where Template:Math is the Template:Mvar-fold loop space of the Template:Mvar-sphere Template:Mvar, and similarly Template:Math is denoted by Template:Math. Therefore the Barratt–Priddy theorem can also be stated as

$B{\mathrm{\Sigma }}_{\mathrm{\infty }}^{+}\simeq {\mathrm{\Omega }}_0^{\mathrm{\infty }}{S}^{\mathrm{\infty }}$ or

$\mathbf{\text{Z}}\times B{\mathrm{\Sigma }}_{\mathrm{\infty }}^{+}\simeq {\mathrm{\Omega }}^{\mathrm{\infty }}{S}^{\mathrm{\infty }}$

In particular, the homotopy groups of Template:Math are the stable homotopy groups of spheres:

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

"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 Template:Math should be the symmetric group Template:Math. The higher K-groups Template:Math of a ring R can be defined as

$K_i(R)={\pi }_i(BG{L}_{\mathrm{\infty }}(R{)}^{+})$

According to this analogy, the K-groups Template:Math of Template:Math should be defined as Template:Math, which by the Barratt–Priddy theorem is:

$K_i({𝐅}_1)={\pi }_i(BG{L}_{\mathrm{\infty }}({𝐅}_1{)}^{+})={\pi }_i(B{\mathrm{\Sigma }}_{\mathrm{\infty }}^{+})={\pi }_i^s.$

References

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