It is proved in Theorem 3.44 of Hatcher's Algebraic Topology (http://pi.math.cornell.edu/~hatcher/AT/ATplain.pdf) that if $K$ is a compact, locally contractible subspace of a closed orientable $n$-manifold $M$, then $H_i(M, M -K;\Bbb Z) = H^{n-i}(K; \Bbb Z)$ for all $i$. In particular, if $M$ is a closed orientable $n$-manifold and $K$ is a compact, codimension $\geq 2$ submanifold of $M$, we can easily see that $H_0(M-K;\Bbb Z)=\Bbb Z$ using the homology long exact sequence for $(M,M-K)$, which implies that $M-K$ is connected and hence path-connected (since $M-K$ is locally path-connected). But I think the assumptions that $M$ being compact, orientable and $K$ being compact are restrictive, and I'm curious about the following questions:
If $M$ is an $n$-manifold and $K$ is a codimension $\geq 2$ submanifold then $M-K$ is path-connected. Is this statement true?
If 1 is true, then is the following more general statement true? If $M$ is a complex $n$-manifold and $K$ is a (complex) codimension $\geq 1$ analytic subvariety, then $M-K$ is path-connected.