Foiled by multiple roots

Let \Sigma_3 be the set of monic complex polynomials of degree 3 with multiple roots, and regard it as a subspace of \mathbb{C}^3 by looking at the non-leading coefficients. Prove that \mathbb{C}^3\setminus \Sigma_3 is homotopy equivalent to S^3\setminus K, where K is the trefoil knot.

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Foiled by multiple roots

Your degree is a chance for expansion

Let X and Y be compact oriented smooth manifolds of the same dimension, with Y connected. Let f : X \to Y be a smooth function. Take y \in Y a regular value of f and, for x \in f^{-1}(y), define o(x) as +1 if f preserves orientation at x and as -1 if it reverses the orientation. We define the degree of f as the number d(f) = \sum_{x\in f^{-1}(y)}o(x).

Prove the following extension theorem: Assume Y=S^k is a k-dimensional sphere and X=\partial W is the boundary of a compact oriented (k+1)-manifold W. There exists a smooth extension F:W\to Y of f if and only if d(f)=0.

Your degree is a chance for expansion