Construct smooth mapping $f: B^{n + 1} \to S^n$ with two singularities at which $f$ has degree $+/- 1$.

Question

I'm currently working through a paper by Pjotr Hajlasz who wants to show that

For smooth manifolds $M,N$, if $\pi_{[p]}(N) \neq 0$ and $1 \leq p < n = \dim M$, then the smooth mappings $C^\infty(M,N)$ are not dense in $W^{1,p}(M,N)$.

In his proof he asserts that

"It is easy to construct a smooth mapping $f: B^{[p]+1} \to S^{[p]}$ with two singularities such that $f$ restricted to small spheres centered at the singularities have degree +1 and -1 respectively."

How would one go about to construct or at least think about such smooth mapping?

I'd appreciate any help!

Answer 1

Use the fact that $B^{n+1}\setminus\{p,q\}\cong S^n\times(0,1)$, and define your map $f\colon S^n\times(0,1)\to S^n$ to just be projection onto the first factor.