Barratt-Priddy-Quillen theorem

The Barratt-Priddy-Quillen theorem is a theorem from stable homotopy theory , a branch of algebraic topology within mathematics . It is named after Michael Barratt , Stewart Priddy, and Daniel Gray Quillen .

statement

Designate the -dimensional sphere and the space of the mappings of degree 0 and with for a fixed point , with the topology of uniform convergence . Further denote the symmetrical group of elements and . ${\displaystyle S^{n}}$ ${\displaystyle n}$ ${\displaystyle Map_{0}(S^{n},S^{n})}$ ${\displaystyle f\colon S^{n}\to S^{n}}$ ${\displaystyle f(x)=x}$ ${\displaystyle x\in S^{n}}$ ${\displaystyle \Sigma _{n}}$ ${\displaystyle n}$ ${\displaystyle \textstyle \Sigma _{\infty }=\bigcup _{n}\Sigma _{n}}$

Barratt-Priddy's theorem says

${\displaystyle H_{k}(\Sigma _{n})=H_{k}(Map_{0}(S^{n},S^{n}))}$

for everyone . ${\displaystyle n\geq 2k}$

An equivalent wording says

${\displaystyle \mathbb {Z} \times B\Sigma _{\infty }^{+}\simeq \Omega ^{\infty }S^{\infty }}$.

Here, the application of the plus construction refers to the classifying space . ${\displaystyle B\Sigma _{\infty }^{+}}$ ${\displaystyle B\Sigma _{\infty }}$

In particular, one obtains the following relationship to the stable homotopy group of the sphere:

${\displaystyle \pi _{i}(B\Sigma _{\infty }^{+})=\pi _{i}^{s}(S^{0})}$

literature

• M. Barratt, S. Priddy: On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici 47, 1-14