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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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Can you clear up the question? Is $M$ orientable? And how do you interpret what happens with an annulus? – Steve D Nov 19 '11 at 16:14
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@SteveD $M$ should be orientable and $1$ means fundamental class, I guess (the answer is "yes" than). – Grigory M Nov 19 '11 at 18:46
Yes, I thought as much, but the OP should state it more carefully. – Steve D Nov 19 '11 at 19:10

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