Non trivial element in the second homotopy group of a manifold

Question

Let $M$ be a closed orientable $n$-dimensional manifold and $\Sigma$ be a $2$-dimensional sphere embedded in $M$ such that there is a map $f:\Sigma\rightarrow \mathbb{S}^2$ with non zero degree, i.e., $deg (f)\not=0$. Is it true that the embedding of $\Sigma$ in $M$ represent a non trivial element of $\pi_2(M)$? if so, how can I prove it?

Answer 1

Let $M$ be $S^3$, The stereotgraphic projection allows to identify $S^3-\{point\}$ to $\mathbb{R}^3$, there are spheres imbedded in $\mathbb{R}^3$ but $\pi_2(S^3)=1$.

Answer 2

I'm guess you mean that there is a map $f:M\to S^2$ whose restriction to $\Sigma$ has non-zero degree. In that case it is true: otherwise there is a homotopy $F:\Sigma \times [0, 1] \to M$ contarcting $\Sigma$ to a point, and composing $F$ with $f$ gives a contracting homotopy from $f|_{\Sigma}=f\cdot F_0: \Sigma \to S^2$ to the constant map $f\cdot F_1: \Sigma \to S^2$