Questions tagged [spanier-whitehead-duality]

The usual $G$-equivariant homology and cohomology groups of a space $X$ with $G$-action are given by the Borel construction: $$H_\ast^G(X)=H_\ast((X\times EG)/G),$$ $$H^\ast_G(X)=H^\ast((X\times EG)/G)...

Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first ...

The question is inspired by an answer to The concept of Duality It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...

Let me first ask an intuitive version of the question: Let $Sp$ be the homotopy category of spectra. Let $E$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we ...

Let $D_1KO$ be the $K(1)$-local Spanier-Whitehead dual of $KO$, i.e. the spectrum $$ D_1KO = F(KO,L_{K(1)}S^0). $$ I am interested in what this is. In fact I know that $D_1KO = \Sigma^{-1} KO$. One ...

Consider a map of spheres $f:S^n\to S^m$ covered by a map of trivial $\mathbb R^k$-bundles. In other words, we take the trivial rank $k$ vector bundle over $S^m$ and pull it to $S^n$ via $f$. Consider ...