$S^n$ is not homotopy equivalent to any subspace $\mathbb{R}^n$ [duplicate]

Solution 1:

By a corollary of the generalized Poincaré duality, referred as Poincaré Alexander Lefschetz duality in the book Geometry and Topology of Bredon, you can derive the following precious tool :

Corollary 8.5 (Bredon - G. & T.) If $L$ is a proper compact subset of an orientable connected $n$-manifold $M$, then $\check H^n(L; G) = 0$ for any coefficient group $G$.

So, if $X$ is a compact subset of $\mathbb R^n$, $X$ can't have the (weak) homotopy type of an $n$-sphere. Now more generally, if $X\subset \mathbb R^n$ is any subset and if it is homotopy equivalent to the $n$-sphere, then you get maps $\mathbb S^n\to X$ and $X \to \mathbb S^n$ such that the compositions are homotopic to the identities. If $L$ is the image of $\mathbb S^n\to X$, then $\mathbb S^n\to L\to \mathbb S^n$ is homotopic to $1_{\mathbb S^n}$ so the maps induced in Cech cohomology verify that $$\check H^n(\mathbb S^n)\to \check H^n(L)\to \check H^n(\mathbb S^n)$$ is the identity, which is not possible because Corollary 8.5 of Bredon applies to $L$ which is compact and so, $\check H^n(L)=0$.