Let be the set of monic complex polynomials of degree 3 with multiple roots, and regard it as a subspace of by looking at the non-leading coefficients. Prove that is homotopy equivalent to , where is the trefoil knot.
Let and be compact oriented smooth manifolds of the same dimension, with connected. Let be a smooth function. Take a regular value of and, for , define as if preserves orientation at and as if it reverses the orientation. We define the degree of as the number .
Prove the following extension theorem: Assume is a -dimensional sphere and is the boundary of a compact oriented -manifold . There exists a smooth extension of if and only if .
Compute the homotopy groups of the compact surfaces.
For an embedded compact 3-manifold , implies .