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. It 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 Template:Math is the topological space of all continuous maps Template:Math from the Template:Mvar-dimensional sphere Template:Mvar to itself, under the topology of uniform convergence (a special case of the compact-open topology). These maps are required to fix a basepoint Template:Math, satisfying Template: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 Template:Mvar.

It follows from the Freudenthal suspension theorem and the Hurewicz theorem that the Template:Mvarth homology Template:Math of this mapping space is independent of the dimension Template:Mvar, as long as Template:Math. Similarly, Nakaoka (1960) proved that the Template:Mvarth group homology Template:Math of the symmetric group Template:Mvar on Template:Mvar elements is independent of Template:Mvar, as long as Template:Math. This is an instance of homological stability.

The Barratt–Priddy theorem states that these "stable homology groups" are the same: for Template:Math 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 Template:Math. The first homology of a group is the largest commutative quotient of that group. For the permutation groups Template:Mvar, the only commutative quotient is given by the sign of a permutation, taking values in Template:Math}. This shows that Template:Math, the cyclic group of order 2, for all Template:Math. (For Template:Math, Template:Math is the trivial group, so Template:Math.)

It follows from the theory of covering spaces that the mapping space Template:Math of the circle Template:Math is contractible, so Template:Math. For the 2-sphere Template:Math, the first homotopy group and first homology group of the mapping space are both infinite cyclic: Template:Math. A generator for this group can be built from the Hopf fibration Template:Math. Finally, once Template:Math, both are cyclic of order 2: Template:Math≅Z/2Z.

Reformulation of the theorem

The infinite symmetric group Template:Math is the union of the finite symmetric groups Template:Math, and Nakaoka's theorem implies that the group homology of Template:Math is the stable homology of Template:Math: for Template:Math, Template:Math. The classifying space of this group is denoted Template:Math, and its homology of this space is the group homology of Template:Math: Template:Math.

We similarly denote by Template:Math the union of the mapping spaces Template:Math (under the inclusions induced by suspension). The homology of Template:Math is the stable homology of the previous mapping spaces: for Template:Math, Template:Math.

There is a natural map Template:Math (one way to construct Template:Mvar is via the model of Template:Math as the space of finite subsets of Template:Math endowed with a certain topology). An equivalent formulation of the Barratt–Priddy theorem is that Template:Mvar is a homology equivalence (or acyclic map), meaning that Template:Mvar 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

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

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

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

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

"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

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

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:

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

References

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