Barratt–Priddy theorem


In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem 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 is the topological space of all continuous maps from the -dimensional sphere to itself, under the topology of uniform convergence. These maps are required to fix a basepoint, satisfying, 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.
It follows from the Freudenthal suspension theorem and the Hurewicz theorem that the th homology of this mapping space is independent of the dimension, as long as. Similarly, proved that the th group homology of the symmetric group on elements is independent of, as long as. This is an instance of homological stability.
The Barratt–Priddy theorem states that these "stable homology groups" are the same: for, there is a natural isomorphism
This isomorphism holds with integral coefficients.

Example: first homology

This isomorphism can be seen explicitly for the first homology. The first homology of a group is the largest commutative quotient of that group. For the permutation groups, the only commutative quotient is given by the sign of a permutation, taking values in. This shows that, the cyclic group of order 2, for all.
It follows from the theory of covering spaces that the mapping space of the circle is contractible, so
. For the 2-sphere, the first homotopy group and first homology group of the mapping space are both infinite cyclic:
A generator for this group can be built from the Hopf fibration. Finally, once, both are cyclic of order 2:

Reformulation of the theorem

The infinite symmetric group is the union of the finite symmetric groups, and Nakaoka's theorem implies that the group homology of is the stable homology of : for,
The classifying space of this group is denoted, and its homology of this space is the group homology of :
We similarly denote by the union of the mapping spaces under the inclusions induced by suspension. The homology of is the stable homology of the previous mapping spaces: for,
There is a natural map ; one way to construct this map is via the model of as the space of finite subsets of endowed with an appropriate topology. An equivalent formulation of the Barratt–Priddy theorem is that is a homology equivalence, meaning that 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.
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
In particular, the homotopy groups of are the stable homotopy groups of spheres:

"''K''-theory of '''F'''1"

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