# Spanier-Whitehead duality

> This is one of the motivations given in Adams' book, which you should read... modulo the part on [smash products](smash%20products).

Given a finite [CW-complex](CW-complex), $X$, I can embed it into a large sphere $S^n$ in a nice way so that the complement deformation retracts onto a finite CW-complex, $Y$. This is called the **Spanier-Whitehead dual** of $X$. 

The thing is, $Y$ is not determined up to homotopy by $X$. However, the [stable homotopy](stable%20homotopy.md) type of $Y$ is determined by $X$, independent of the choice of embedding or sphere.
This basically follows from [Alexander duality](Alexander%20duality.md). 
So if we want to make arguments exploiting duality, it's best to work in the stable category.