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Let $(M,g)$ be a spin manifold with boundary $\partial M= N_1 \cup N_2$, $N_i$ connected spin manifolds. Suppose that $H^*(M,\partial M)=0$ and the index of the Dirac operator of $M$, $ind_M(\mathcal{D} )=0$, then have we:

$$ind_{N_1}(\mathcal{D})=ind_{N_2}(\mathcal{D})$$

?

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