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:

$$H_i(M \setminus p) \to H_i(M)$$

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 $$i \ne n$$, then the long exact sequence of homology of a pair $$(M, M \setminus p)$$ yields that for $$i \ne n, n -1$$, 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 $$i \ne n, n - 1$$. 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

See also: multiple point-deletion inclusion, submanifold-deletion inclusion