Degree one normal maps to the 3-sphere (Ex)

Let

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$<wikitex>; Let $M$ be a closed oriented 3-manifold. # Prove that $M$ that there is a degree one normal map $(f, \overline f) \colon M \to S^3$ (since $\pi_3(BSO) = 0$, the target bundle is trivial). # Prove that $(f, \overline f) \colon M \to S^3$ is normally bordant to a homotopy equivalence. </wikitex> [[Category:Exercises]] [[Category:Exercises without solution]]M$ be a closed oriented 3-manifold.

1. Prove that

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   $M$ that there is a degree one normal map $(f, \overline f) \colon M \to S^3$$(f, \overline f) \colon M \to S^3$ (since $\pi_3(BSO) = 0$$\pi_3(BSO) = 0$, the target bundle is trivial).
2. Prove that $(f, \overline f) \colon M \to S^3$$(f, \overline f) \colon M \to S^3$ is normally bordant to a homotopy equivalence.