The author obtains new general topological properties of complements of a holomorphically convex subset \(A \subset X\) in a Stein manifold \(X\) of dimension \(n \geq 2\). Let us formulate one of his results.
Theorem 1. If \(A \subset \mathbb{C}^ n\) \((n \geq 2)\) is either a polynomially convex of \(\mathbb{C}^ n\) or the closure \(A = \overline{\Omega}\) of a bounded pseudoconvex Runge domain \(\Omega \subset \mathbb{C}^ n\) with \(C^ 1\) boundary, then the complement of \(A\) is \((n - 1)\)-connected: \[ \pi_ k(\mathbb{C}^ n \setminus A) = 0,\quad 1 \leq k \leq n - 1, \] and for any abelian group \(G\) \[ H_ k(\mathbb{C}^ n \setminus A; G) = 0,\quad 1 \leq k \leq n - 1. \]