Adjacent connected sums and torus actions

Abstract:

Let M and N be closed, compact manifolds of dimension m and let X be a closed manifold of dimension $n < m$ with embeddings of $X \times {D^{m - n}}$ into M and N. Suppose the interior of $X \times {D^{m - n}}$ is removed from M and N and the resulting manifolds are attached via a homeomorphism $f: X \times {S^{m - n - 1}} \to X \times {S^{m - n - 1}}$. Let this homeomorphism be of the form $f(x, t) = (x, F(x)(t))$ where $F: X \to SO(m - n)$. The resulting manifold, written as $M {\# _X} N$, is called the adjacent connected sum of M and N along X. In this paper definitions and examples are given and the examples are then used to classify actions of the torus ${T^n}$ on closed, compact, connected, simply connected $(n + 2)$-manifolds, $n \geqslant 4$.