Consider $M,N$ submanifolds of some manifold $X$ such that $\dim M+\dim N=\dim X$. For $x\in M\cap N$, let $\langle M_{x},N_{x}\rangle$ denote the index by matching local orientation of $TM_{x},TN_{x}$ together with that of $X$. Define $[\partial N, M]$ to be the sum of the indices. If $\partial M\cap \partial N=\emptyset$, prove $$[\partial M, N]=[\partial N, M]$$

My professor give a 2-minute hand waving proof using cup product and pairing between cohomology and homology ($\langle M,N\rangle=[M]\cup [N]$). I am wondering if there are simpler proofs, for this should be easy and intuitive. I could not find a proof myself.


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