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 $M ∖ p$ in $M$ viz the homomorphisms:

$H_{i} (M ∖ p) \to H_{i} (M)$

Effect on homology

For a homologically Euclidean point

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

For a manifold

If $M$ is a manifold of dimension $n$ , then every point $p$ is homologically Euclidean, so $H_{i} (M ∖ p) \to H_{i} (M)$ 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:

$H_{n} (M) \to H_{n} (M, M ∖ p)$

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

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

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

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

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

Effect on fundamental group

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