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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 1

### Splittings in the Burnside ring and in $SF_G$

Pages: 1 – 27

DOI: http://dx.doi.org/10.4310/HHA.2008.v10.n1.a1

#### Author

#### Abstract

Let $G$ be a finite $p$-group, $p \neq 2$. We construct a map from the space $J_G$, defined as the fiber of $\psi^k-1: B_G O \to B_G Spin$, to the space $(SF_G)_p$, defined as the 1-component of the zeroth space of the equivariant $p$-complete sphere spectrum. Our map produces the same splitting of the $G$-connected cover of $(SF_G)_p$ as we have described in previous work, but it also induces a natural splitting of the $p$-completions of the component groups of fixed point subspaces.

#### Keywords

$J$-homomorphism; Burnside ring; sphere spectru

#### 2010 Mathematics Subject Classification

19L20, 19L47, 55R91