Barratt–Priddy 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 -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 th homology H_k(\operatorname{Map}_0(S^n,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 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(\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 {{math|{−1, 1}}}. 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}_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 \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(\operatorname{Map}_0(S^n,S^n))=H_1(\operatorname{Map}_0(S^n,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 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 \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_\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 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 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_\infty(R)^+) According to this analogy, the K-groups of should be defined as , 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==