Barratt–Priddy theorem explained

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} ^{n,S} _{0 (S} ^{n)} is the topological space of all continuous maps f\colon S ^{n} \to S ^{n} from the -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} .H _{k(\operatorname{Map}} ^{n,S} ^{0 (S} ^{n))} of this mapping space is independent of the dimension, as long as n>k . Similarly, proved that the th group homology H _{k(\Sigma} _{n)} of the symmetric group \Sigma _{n} on elements is independent of, as long as n \ge 2 k . This is an instance of homological stability.

The Barratt–Priddy theorem states that these "stable homology groups" are the same: for

n \ge 2 k , there is a natural isomorphismH _{k(\Sigma} _{n)\cong} H _{k(Map} ^{n,S} ^{0 (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 . 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} ^{1 ,S} _{0 (S} ^{1)} of the circle S ^{1} is contractible, soH _{1(\operatorname{Map}} ^{1 ,S} ^{0 (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}} ^{2 ,S} ^{0 (S} ^{2 )) = H} ^{1 (\operatorname{Map}} ^{2 ,S} ^{0 (S} ^{2))\cong} \Z . S ^{3} \to S ^{2} . Finally, once n\ge 3 , both are cyclic of order 2:\pi _{1(\operatorname{Map}} ^{n,S} ^{0 (S} ^{n)) = H} ^{1 (\operatorname{Map}} ^{n,S} ^{0 (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 2 k , 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} ^{inf ty} _{0 (S} ,S ^{infty} ) the union of the mapping spaces \operatorname{Map} ^{n} _{0 (S} ,S ^{n} ) under the inclusions induced by suspension. The homology of \operatorname{Map} ^{inf ty} _{0 (S} ,S ^{infty} ) is the stable homology of the previous mapping spaces: for n>k , H _{k(\operatorname{Map}} ^{inf ty} ^{0 (S} ,S ^{infty} )) \cong H _{k(\operatorname{Map}} ^{n} ^{0 (S} ,S ^{n} )).

There is a natural map

\varphi\colon B\Sigma _{infty} \to \operatorname{Map} ^{inf ty} _{0 (S} ,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 resulting from applying Quillen's plus construction to can be identified with . (Since, the map satisfies the universal property of the plus construction once it is known that is a homology equivalence.)

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

B\Sigma ^{+ \simeq} _{inf ty} \Omega ^{inf ty} _{0} S ^{infty} orbf{Z} x B\Sigma ^{+ \simeq} _{inf ty} \Omega ^{infty} S ^{infty}

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

\pi _{i(B\Sigma} ^{+ )\cong} ^{inf ty} \pi ^{inf ty} _{i(\Omega} S ^{infty)\cong} \lim _{n →} \pi _{n+i} (S ^{s(S} _{i} ^{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 should be the symmetric group .The higher K-groups of a ring R can be defined as

K _{i(R)=\pi} _{i(BGL} ^{+ )} ^{inf ty(R)}

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

K _{i(F} _{1)=\pi} _{i(BGL} _{infty(F} ^{+ ) = \pi} ^{i(B\Sigma} ^{s.} ^{i}