nLab n-duality

Idea

The fact that for a manifold $X$ of dimension $k$, then for any embedding $\iota \colon X\hookrightarrow \mathbb{R}^n$ (which exists by the Whitney embedding theorem) the Thom space $X^{\nu}$ of the normal bundle $\nu$ behaves like a dual to $X_+$ under smash product of pointed topological spaces.

Under passing to suspension spectra this becomes Atiyah duality in stable homotopy theory.

Related concepts

• Atiyah duality

Referenes

• Cary Malkiewich, p.15 of The stable homotopy category, 2014 (pdf)

For ex-spaces see around def. 1.4.2 of

• Kate Ponto, Fixed point theory and trace for bicategories (pdf)

For parametrized spectra see page xyz of

• Peter May, J. Sigurdsson, Parametrized Homotopy Theory