In a sense, the reason that the duality fails is that near the intersection of the two great circles, the set of boundary points where the forms are allowed to be non-zero is disconnected, and that no matter how small a neighbourhood we choose in $B^3$ for the intersection point, its cohomology will therefore not be entirely elementary. We This can prevent this be prevented by demanding that any every point in $\partial M$ has a an "elementary" neighbourhood where $U \cong \mathbb{H}^{n}$ such that
• the division subdivision of $\partial U$ into faces looks like is diffeomorphic to a small neighbourhood in the boundary complete fan (a subdivision of an $n$-simplex (in other words\mathbb{R}^{n-1}$into simplicial cones), •$V$has a basis $\{e_i\}$ such that for each face$F$meeting$U$,$V_F$is spanned by a subset, • for each$e_i$, the product interior in$U$of the boundary union of an open$k$-simplex and the faces$\mathbb{R}^{n-k}$for some F$ such that $k$); e.ge_i \not\in V_F$is connected.only 3 faces are allowed to meet at an edge • Essentially, 1. says that the subdivision of codimension 2 in$\partial M$(while is sensible, 3. prevents situations like in the example 4 faces meet a vertex)above, and 2. makes sure we can state 3. sensibly when$\dim V > 1$(see example in Trial's comment below).I think that if$M^n$is oriented with boundary and every point in$\partial M$has possesses such a "simplex boundary" neighbourhood for the subdivisionelementary" neighbourhoods, thenwhere the subscript$c$indicates the cohomology of a complex with compact supports. It should be possible to prove this using induction on a good cover (and the duality between the Mayer-Vietoris sequences for normal and compactly supported de Rham cohomology) like for standard Poincaré duality, provided that the statement is true for open subsets$U \subset M$diffeomorphic to$\mathbb{R}^n$or and for the half-space$\mathbb{H}^n$."elementary" neighbourhoods. For$U \cong \mathbb{R}^n$this is just usual Poincaré duality tensored with$V$. For$U \cong \mathbb{H}^n$, we can restrict to the case where$\partial U$is a "simplex boundaryan "elementary" neighbourhood for the subdivision of$\partial M$. If we let$V_{\partial U}$denote the intersection of$V_F$over all faces$F$meeting$\partial U$, then U$,$H^0_{V,$H^k_{V, \{V_F\}}(U) = \cong V_{\partial U}$ while integration gives a well-defined map $H^n_{c, V^*bigoplus_i H^{k}_{V_i, \{ \text{ann} V_F\}}(U)^* \to V^*/\text{ann}V_{\partial U} {V_F \cong V_{\partial U}^*$, which is an isomorphism. Checking that the other cohomology groups (with boundary conditionscap V_i\}}(U) of$U$vanish is easy when$\dim V = 1$(for $H^{n-1}_{c, $H^k_{c, V^*, \{ {\text{ann} V_F\}}(U) = \text{ann} V_F\}}(U)$ we use that the interior of the union of the faces bigoplus_i H^{k}_{c, V_i^*, \{\text{ann} (V_F \cap V_i) \}}(U), $F$ in $\partial U$ with where $V_F = 0$ is connected, which V_i$is where the "simplex boundary" condition comes in), and span of the clearest way to prove it general seems to be by induction on element$\dim V$. For the induction step, use e_i$ of the exact sequence basis from condition 2.The terms on the short exact sequence$right hand side all vanish, except that if$0 e_i \to\Omega^*_{W, in V_F$for all$F$meeting$\partial U$then$H^0_{V_i, \{ V_F {V_F \cap W\}}(U) \to \Omega^*_{V, \{ V_F\}V_i\}} \to \Omega^*_{V/W, cong V_i$and$H^{n}_{c, V_i^*, \{ V_F/W {\text{ann} (V_F \}}(U) cap V_i)\}}(U) \cong V_i^*$(3. is used to 0$$where show that W H^{n-1}_{c, V_i^*, \subset V is any subspace {\text{ann} (and V_F \cap V_i)\}}(U) = 0). So the analogous sequence duality holds for complexes with compact support)the "elementary" neighbourhoods. 1 An example where the duality fails is when M^n is the closed unit ball B^3 \subset \mathbb{R}^3, and its boundary S^2 is divided into four quarters by 2 great circles. If V = \mathbb{R}, V_F = V for 2 opposite quarters F and V_F = 0 for the other two, then H^1_{V, \{ V_F \}}(M) = 0 while H^2_{V^*, \{\text{ann} V_F\}} \cong \mathbb{R} (essentially, they are H^1_c and H^2_c, respectively, of the product of an open 2-disc and a closed interval). In a sense, the reason that the duality fails is that near the intersection of the two great circles, the set of boundary points where the forms are allowed to be non-zero is disconnected, and that no matter how small a neighbourhood we choose in B^3 for the intersection point, its cohomology will therefore not be entirely elementary. We can prevent this by demanding that any point in \partial M has a neighbourhood where the division into faces looks like a small neighbourhood in the boundary of an n-simplex (in other words, the product of the boundary of an open k-simplex and \mathbb{R}^{n-k} for some k); e.g. only 3 faces are allowed to meet at an edge of codimension 2 in \partial M (while in the example 4 faces meet a vertex). I think that if M^n is oriented with boundary and every point in \partial M has such a "simplex boundary" neighbourhood for the subdivision, then $$H^k_{V, \{V_F\}}(M) \cong H^{n-k}_{c, V^*, \{ \text{ann} V_F\}}(M)^*$$ where the subscript c indicates the cohomology of a complex with compact supports. It should be possible to prove this using induction on a good cover (and the duality between the Mayer-Vietoris sequences for normal and compactly supported de Rham cohomology) like for standard Poincaré duality, provided that the statement is true for open subsets U \subset M diffeomorphic to \mathbb{R}^n or the half-space \mathbb{H}^n. For U \cong \mathbb{R}^n this is just usual Poincaré duality tensored with V. For U \cong \mathbb{H}^n, we can restrict to the case where \partial U is a "simplex boundary" neighbourhood for the subdivision of \partial M. If we let V_{\partial U} denote the intersection of V_F over all faces F meeting \partial U, then H^0_{V, \{V_F\}}(U) \cong V_{\partial U} while integration gives a well-defined map H^n_{c, V^*, \{ \text{ann} V_F\}}(U)^* \to V^*/\text{ann}V_{\partial U} \cong V_{\partial U}^*, which is an isomorphism. Checking that the other cohomology groups (with boundary conditions) of U vanish is easy when \dim V = 1 (for H^{n-1}_{c, V^*, \{ \text{ann} V_F\}}(U) we use that the interior of the union of the faces F in \partial U with V_F = 0 is connected, which is where the "simplex boundary" condition comes in), and the clearest way to prove it general seems to be by induction on \dim V. For the induction step, use the exact sequence from the short exact sequence $$ 0 \to\Omega^*_{W, \{ V_F \cap W\}}(U) \to \Omega^*_{V, \{ V_F\}} \to \Omega^*_{V/W, \{ V_F/W \}}(U) \to 0$$ where$W \subset V\$ is any subspace (and the analogous sequence for complexes with compact support).