Point-deletion inclusion

Definition

Let $M$ be a topological space and $p\in M$ a (closed) point. The point-deletion inclusion problem studies the map induced on homology, fundamental groups and other homotopy invariants by the inclusion of $M\setminus p$ in $M$ viz the homomorphisms:

$H_{i}(M\setminus p)\to H_{i}(M)$

Effect on homology

For a homologically Euclidean point

If $p$ is a $n$ -homologically Euclidean point viz if $H_{n}(M,M\setminus p)=\mathbb {Z}$ and $H_{i}(M,M\setminus p)=0$ for $i\neq n$ , then the long exact sequence of homology of a pair $(M,M\setminus p)$ yields that for $i\neq n,n-1$ , the inclusion induces an isomorphism on $i^{th}$ homology.

For a manifold

If $M$ is a manifold of dimension $n$ , then every point $p$ is homologically Euclidean, so $H_{i}(M\setminus p)\to H_{i}(M)$ is an isomorphism for $i\neq n,n-1$ . What happens at $n,n-1$ depends on the nature of the manifold.

If $M$ is a compact connected orientable manifold, then the map:

$H_{n}(M)\to H_{n}(M,M\setminus p)$

is an isomorphism, and hence we see that $H_{n}(M\setminus p)=0$ , while the map $H_{n-1}(M\setminus p)\to H_{n-1}(M)$ is an isomorphism.

If $H_{n}(M)=0$ (which could occur if $M$ is compact non-orientable, or where $M$ is non-compact), then $H_{n}(M\setminus p)=0$ , and we get:

$H_{n-1}(M\setminus p)=H_{n-1}(M)\oplus H_{n}(M,M\setminus p)=H_{n-1}(M)\oplus \mathbb {Z}$

(the proof of this relies on getting a splitting of a short exact sequence; the splitting is not canonical.

Note that in both cases, $H_{n}(M\setminus p)=0$ ; but in the second case, the map is an isomorphism on $n^{th}$ homology while n the first case the map is an isomorphism on $(n-1)^{th}$ homology.

Effect on fundamental group

Point-deletion inclusion induces isomorphism on fundamental groups for manifold of dimension at least two