Barratt–Priddy theorem

Short description: Connects the homology of the symmetric groups with mapping spaces of spheres

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 [math]\displaystyle{ \operatorname{Map}_0(S^n,S^n) }[/math] is the topological space of all continuous maps [math]\displaystyle{ f\colon S^n \to S^n }[/math] from the n-dimensional sphere [math]\displaystyle{ S^n }[/math] to itself, under the topology of uniform convergence (a special case of the compact-open topology). These maps are required to fix a basepoint [math]\displaystyle{ x\in S^n }[/math], satisfying [math]\displaystyle{ f(x)=x }[/math], 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 [math]\displaystyle{ \Sigma_n }[/math].

It follows from the Freudenthal suspension theorem and the Hurewicz theorem that the kth homology [math]\displaystyle{ H_k(\operatorname{Map}_0(S^n,S^n)) }[/math] of this mapping space is independent of the dimension n, as long as [math]\displaystyle{ n\gt k }[/math]. Similarly, Minoru Nakaoka (1960) proved that the kth group homology [math]\displaystyle{ H_k(\Sigma_n) }[/math] of the symmetric group [math]\displaystyle{ \Sigma_n }[/math] on n elements is independent of n, as long as [math]\displaystyle{ n \ge 2k }[/math]. This is an instance of homological stability.

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

[math]\displaystyle{ H_k(\Sigma_n)\cong H_k(\text{Map}_0(S^n,S^n)). }[/math]

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 [math]\displaystyle{ H_1 }[/math]. The first homology of a group is the largest commutative quotient of that group. For the permutation groups [math]\displaystyle{ \Sigma_n }[/math], the only commutative quotient is given by the sign of a permutation, taking values in {−1, 1}. This shows that [math]\displaystyle{ H_1(\Sigma_n) \cong \Z/2\Z }[/math], the cyclic group of order 2, for all [math]\displaystyle{ n\ge 2 }[/math]. (For [math]\displaystyle{ n= 1 }[/math], [math]\displaystyle{ \Sigma_1 }[/math] is the trivial group, so [math]\displaystyle{ H_1(\Sigma_1) = 0 }[/math].)

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

[math]\displaystyle{ \pi_1(\operatorname{Map}_0(S^2,S^2))=H_1(\operatorname{Map}_0(S^2,S^2))\cong \Z }[/math].

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

[math]\displaystyle{ \pi_1(\operatorname{Map}_0(S^n,S^n))=H_1(\operatorname{Map}_0(S^n,S^n))\cong \Z/2\Z }[/math].

Reformulation of the theorem

The infinite symmetric group [math]\displaystyle{ \Sigma_{\infty} }[/math] is the union of the finite symmetric groups [math]\displaystyle{ \Sigma_{n} }[/math], and Nakaoka's theorem implies that the group homology of [math]\displaystyle{ \Sigma_{\infty} }[/math] is the stable homology of [math]\displaystyle{ \Sigma_{n} }[/math]: for [math]\displaystyle{ n\ge 2k }[/math],

[math]\displaystyle{ H_k(\Sigma_{\infty}) \cong H_k(\Sigma_{n}) }[/math].

The classifying space of this group is denoted [math]\displaystyle{ B \Sigma_{\infty} }[/math], and its homology of this space is the group homology of [math]\displaystyle{ \Sigma_{\infty} }[/math]:

[math]\displaystyle{ H_k(B \Sigma_{\infty})\cong H_k(\Sigma_{\infty}) }[/math].

We similarly denote by [math]\displaystyle{ \operatorname{Map}_0(S^{\infty},S^{\infty}) }[/math] the union of the mapping spaces [math]\displaystyle{ \operatorname{Map}_0(S^{n},S^{n}) }[/math] under the inclusions induced by suspension. The homology of [math]\displaystyle{ \operatorname{Map}_0(S^{\infty},S^{\infty}) }[/math] is the stable homology of the previous mapping spaces: for [math]\displaystyle{ n\gt k }[/math],

[math]\displaystyle{ H_k(\operatorname{Map}_0(S^{\infty},S^{\infty})) \cong H_k(\operatorname{Map}_0(S^{n},S^{n})). }[/math]

There is a natural map [math]\displaystyle{ \varphi\colon B\Sigma_{\infty} \to \operatorname{Map}_0(S^{\infty},S^{\infty}) }[/math]; one way to construct this map is via the model of [math]\displaystyle{ B\Sigma_{\infty} }[/math] as the space of finite subsets of [math]\displaystyle{ \R^{\infty} }[/math] endowed with an appropriate topology. An equivalent formulation of the Barratt–Priddy theorem is that [math]\displaystyle{ \varphi }[/math] is a homology equivalence (or acyclic map), meaning that [math]\displaystyle{ \varphi }[/math] 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

[math]\displaystyle{ B\Sigma_\infty^+\simeq \Omega_0^\infty S^\infty }[/math] or

[math]\displaystyle{ \textbf{Z}\times B\Sigma_\infty^+\simeq \Omega^\infty S^\infty }[/math]

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

[math]\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(S^n) }[/math]

"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

[math]\displaystyle{ K_i(R)=\pi_i(BGL_\infty(R)^+) }[/math]

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:

[math]\displaystyle{ K_i(\mathbf{F}_1)=\pi_i(BGL_\infty(\mathbf{F}_1)^+)=\pi_i(B\Sigma_\infty^+)=\pi_i^s. }[/math]

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, http://www.digizeitschriften.de/dms/img/?PPN=PPN358147735_0047&DMDID=dmdlog5 
• Nakaoka, Minoru (1960), "Decomposition theorem for homology groups of symmetric groups", Annals of Mathematics 71 (1): 16–42, doi:10.2307/1969878