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I have an exercise as follows: "Let $M\subset \mathbb{R}^k$ be a smooth, connected, oriented, compact manifold without boundary of dimension $p$. Let $\Omega^{Fr}_n(M)$ be the set of all equivalence classes of framed cobordism of submanifolds of $M$ of dimension $n$. Prove that $\Omega^{Fr}_0(M)=\mathbb{Z}$. Hence we can deduce the homotopy group $\Pi[M,S^p]=\mathbb{Z}$ by the Pontryagin theorem."

I have no idea to associate each equivalence class in $\Omega^{Fr}_0(M)$ with an integer. Someone can help me? Thanks a lot!

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By compactness, the $0$ dimension submanifolds are just finite sets of points. Orientation is going to introduce the sign... – Berci Nov 7 '12 at 15:21

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