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For an orientable n-manifold $M$ with boundary, we know that there is a Lefschetz theorem: $H^{k}(M)\cong(H^{n-k}(M,\partial M))^*$. Now if we consider a representation $\rho$ of the fundamental group of $M$ (for simplicity, we can assume that $\rho$ is trivial when restricted to the boundary). Is there a similar relationship between $H^{k}(M;\rho)$ and $H^{n-k}(M,\partial M;\rho)$?

In three dimensional topology, there is a famous "half die half alive" lemma. Which says when we map the first q-coefficient homology group of the boundary into the 3-manifold, the rank of the kernel is the genus of the boundary. This is a consequence of Lefschetz Duality. Is there a similar theorem when we consider twisted coefficients? (Again, we can assume that the representation is trivial when restrict to the boundary).

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The form of Poincaré duality with twisted or local coefficients that I am familiar with would say that $$H^i(X,\rho)\cong H_c^{n-k}(X,\rho^\vee)^*$$ if we disregard torsion. Here $X$ is an orientable $n$-manifold, and $\rho^\vee$ is the dual or contragredient representation. Taking $X= M-\partial M$ should give the correct form of the statement you want.An overly complicated reference would be Iversen's Cohomology of Sheaves. Look up the more general Verdier dually and specialize to the above case. Alternatively, you can try to modify your favorite existing proof. – Donu Arapura Dec 1 '12 at 13:41

Poincaré-Lefschetz duality for twisted coefficients is fundamental to surgery theory, and for the `universal' case of $Z[\pi_1(X)]$ coefficients is discussed in Chapter 2 of Wall's book, Surgery on Compact Manifolds. As in the case of integral coefficients, the "half die half alive" principle (suitably phrased) holds in arbitrary dimensions.

For the setting which seems to interest you, ie duality with coefficients in a representation, you might look at Milnor's "A duality theorem for Reidemeister torsion", Annals 76 (1962), or alternately his survey "Whitehead torsion" (BAMS 72, 1966). There, the "half die half alive" principle shows up (written in a multiplicative way) as the statement that the torsion of a boundary factors as the torsion of the manifold times its conjugate (in a certain sense). A nice application is the famous Fox-Milnor condition on the Alexander polynomial of a slice knot. This approach using duality with twisted coefficients was greatly extended to the setting of the so-called twisted torsion by Kirk and Livingston (two papers in Topology Vol. 38, pp. 635-661 and 663-671, 1999).

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The Lefschetz duality is more naturally expressed as a duality between cohomology and homology. An oriented $n$-manifold with boundary has a fundamental class $[M]\in H_n(M,\partial M;\mathbb{Z})$, and cap product with this class induces an isomorphism $$ H^k(M;\rho)\stackrel{\cong}{\longrightarrow} H_{n-k}(M,\partial M;\rho\otimes\mathbb{Z})\cong H_{n-k}(M,\partial M;\rho) $$ for any representation $\rho$ of $\pi_1(M)$. Unfortunately the relationship between cohomology and homology with local coefficients is more complicated than with trivial coefficients (due to the Universal Coefficient Theorem not holding in full generality), and so it may be that the homology may not be dual to cohomology (modulo torsion).

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Mark -- shouldn't it be $\rho$ dual on the right hand side? – algori Dec 1 '12 at 15:01
@algori: I don't think so, see – Mark Grant Dec 2 '12 at 17:50

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