Homology of complement of compact submanifold

Statement

Suppose $M$ is a manifold and $K$ is a compact connected manifold embedded as a submanifold inside $M$ . The general problem asks for computation of the homology of $M ∖ K$ , without any prior information about how $K$ has been embedded inside $M$ .

Particular cases

In general, the answer is not determined, and depends on the way that $K$ sits inside $M$ . However, for certain special cases, the answer can be determined; for instance, when $K$ is a single point, is the problem of point-deletion inclusion, where the answer is completely determined in terms of the homology of $M$ , except at dimensions $n$ and $n - 1$ .

When the submanifold is a sphere

Suppose $M$ has dimension $n$ and $K = S^{m}$ has dimension $m$ . Then the Alexander duality theorem yields that:

$H_{n} (M, M ∖ K) = H_{n - m} (M, M ∖ K) = Z$

and all other relative homologies are 0. This puts significant constraints on the homologies of $M ∖ K$ ; nonetheless these homologies are not determined uniquely by the data.

When the manifold is compact connected orientable

If $M$ itself is a compact connected orientable manifold, and $K$ is any compact connected manifold, then we have:

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

All three groups are $Z$ and the composite is an isomorphism; this forces the map from $H_{n} (M)$ to $H_{n} (M, M ∖ K)$ to be zero. Writing down the long exact sequence of homology of a pair $(M, M ∖ K)$ , we see that $H_{n} (M ∖ K) = 0$ , and the long exact sequence can be truncated to begin at:

$0 \to H_{n - 1} (M ∖ K) \to H_{n - 1} (M) \to \dots$

When the manifold and submanifold are spheres

A special case of both the above occurs when $M = S^{n}$ and $K$ is homeomorphic to $S^{m}$ , $m < n$ . In this case, the homologies of the complement of $K$ in $M$ can be determined completely, using the two observations above, and the fact that all lower reduced homologies of $M$ are zero.

What we deduce is that the homologies of the complement are $Z$ at $n - m - 1$ and $0$ if $m \neq n - 1$ , and $Z \oplus Z$ at $0$ if $m = n - 1$ . Homologies are 0 elsewhere.