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If $M$ is a compact orientable manifold with boundary, is it necessary that in the long exact sequence ${H_n}(M,\partial M)\mathop \to \limits^{{\partial _ * }} {H_{n - 1}}(\partial M)$, ${{\partial _ * }}$ sends $1$ to $+1$ or $-1$ in ${H_{n - 1}}(\partial M)$ ?

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    Yes, I thought as much, but the OP should state it more carefully.2011-11-19

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