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.
is the topological space of all continuous maps
These maps are required to fix a basepoint
, 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
of this mapping space is independent of the dimension n, as long as
Similarly, Minoru Nakaoka (1960) proved that the kth group homology
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 (in fact with any coefficients, as is made clear in the reformulation below).
This isomorphism can be seen explicitly for the first homology
, the only commutative quotient is given by the sign of a permutation, taking values in {−1, 1}.
, the cyclic group of order 2, for all
, 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
, both are cyclic of order 2: The infinite symmetric group
is the union of the finite symmetric groups
, and Nakaoka's theorem implies that the group homology of
, The classifying space of this group is denoted
is the stable homology of the previous mapping spaces: for
is a homology equivalence (or acyclic map), meaning that
induces an isomorphism on all homology groups with any local coefficient system.
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 In particular, the homotopy groups of BΣ∞+ are the stable homotopy groups of spheres: 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 GLn(F1) should be the symmetric group Σn.
The higher K-groups Ki(R) of a ring R can be defined as According to this analogy, the K-groups Ki(F1) of F1 should be defined as πi(BGL∞(F1)+)=πi(BΣ∞+), which by the Barratt–Priddy theorem is: