turkmath.org

Given a Poincaré complex $X$, a vector bundle $\xi$ over $X$ is said to be realized by the normal bundle of a manifold $M$, if $\xi$ is pulled back from the normal bundle of $M$ along a homotopy equivalence $X\rightarrow M$. The problem of determining such bundles over an arbitrary Poincaré complex is a difficult problem and is related to classical problems of surgery theory. In this talk, we will discuss some methods of approaching this problem and talk about solutions for certain cases of $X$. In particular, we will discuss conditions on bundles over $X$ that guarantee they are realized by normal bundles of manifolds, for $X$ belonging to a certain class of homology spheres.