Surgery theory

Summary

Surgery is an operation that cuts out a thickened sphere $S^p \times D^q$ from a manifold and glues in $D^{p+1} \times S^{q-1}$ along the common boundary. The result is a new manifold, cobordant to the old one. Surgery theory uses this operation to decide whether a space is homotopy equivalent to a manifold. It also classifies the manifolds within a homotopy type. In dimensions at least $5$ it reduces these questions to homotopy theory and to algebra in the surgery obstruction groups.