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The integral Poincare duality theorem says that a closed connected oriented $n$-manifold has a fundamental class $[M]\in H_n(M;\mathbb{Z})$ for which cap product gives $H^q(M;\mathbb{Z})\cong H_{n-q}(M;\mathbb{Z})$. The general Poincare duality with the orientation local system uses $[M]\in H_n(M;{\mathcal{O}}_M)$ and givesfor a local coefficient system $\mathcal{A}$. Over ${\mathbb{F}}_2$ it holds without orientability. For constant integral coefficients on a nonorientable connected closed manifold, $H_n=H^0(M;{\mathcal{O}}_M)=0$, while $H^n=H_0(M;{\mathcal{O}}_M)=\mathbb{Z}/2$: the invariants of sign reversal in $\mathbb{Z}$ are zero and its coinvariants are $\mathbb{Z}/(a\sim -a)$.

If the three-manifold is orientable, duality and the universal coefficient theorem for cohomology giveFor a nonorientable three-manifold, $H_3=0$ and $H^3=\mathbb{Z}/2$. The top-degree universal coefficient sequence therefore givesSince the homology groups are finitely generated, their cyclic decomposition shows that this Ext group is isomorphic to the torsion subgroup of $H_2$. Thus the torsion is exactly $\mathbb{Z}/2$, not the possibly larger torsion subgroup of $H_1$. Also odd-dimensional closed manifolds have zero Euler characteristic, by pairing complementary mod-two Betti numbers. If $r$ is the free rank of $H_2$, it follows that $0=\chi (M)=1-b+r$, giving the second homology of a closed nonorientable three-manifoldThe inequality is forced by the nonnegative free rank $b-1$.

For the final graded-group construction, one always has $K^0=K^3=K^6=\mathbb{Z}$, since each of these is copied from $H^0$ of a connected three-manifold. Suppose it were the cohomology of a closed six-manifold $N$. The group $K^0$ makes $N$ connected, and $K^6=\mathbb{Z}$ makes it orientable: the nonorientable alternative has top integral cohomology $\mathbb{Z}/2$, as above. Its third rational cohomology would then be one-dimensional. But the middle Poincare duality pairing on $H^3(N;\mathbb{Q})$ is nondegenerate and skew-symmetric by graded commutativity, hence must have even dimension. Equivalently, a generator $u$ would satisfy $u⌣u=-u⌣u$, so its square in the torsion-free top group is zero and the entire rank-one pairing vanishes. This contradicts duality and the even rank of middle cohomology in dimension four k plus two. ThereforeBoth existence questions have negative answers, independently of the other groups of the original three-manifold.

For a Serre fibration $F\to E\to B$ over a path-connected CW complex, the cohomological Serre spectral sequence starts withThe fibre cohomology groups form a local coefficient system ${\mathcal{H}}^q(F;R)$ through the fundamental group action. For a simply connected base, it is constant. The stable terms are successive quotients of a finite filtration of each $H^n(E;R)$; reconstructing the cohomology groups can require solving group extensions. With commutative ring coefficients, the cup product makes the construction multiplicative.^[ref]

For the Hopf fibration $S^1\to S^3\to S^2$ with integer coefficients, the only nonzero positions on $E_2$ are $(0,0),(0,1),(2,0),(2,1)$, each a copy of $\mathbb{Z}$. Since $H^1(S^3)=H^2(S^3)=0$, the map $d_2:E_2^{0,1}\to E_2^{2,0}$ must be an isomorphism. The survivors at $(0,0)$ and $(2,1)$ give the cohomology in degrees zero and three. This example shows how twisting changes the answer from the cohomology of a Cartesian product.

For a Serre fibration $F\to E\to S^n$ with $n\ge 2$, the homological Serre spectral sequence has only columns $0,n$, and its only potentially nonzero differential isUsing an oriented fundamental class of $S^n$ identifies the source and target with $H_i(F)$ and $H_{i+n-1}(F)$. This gives the Wang homomorphism $\Delta$, and the resulting kernel/cokernel filtration pieces splice into the Wang sequence over a sphere.

For a fibration over the circle, the degree-preserving boundary map is instead the monodromy action minus the identity, $T_{\ast }-1:H_i(F)\to H_i(F)$, up to the chosen orientation sign. It arises from the cellular boundary with the fibre-homology local coefficient system. It is already incorporated in the $E_2$ page; there is no additional $d_1$ differential on an $E_2$-starting Serre spectral sequence. The distinction is visible for a product over $S^1$: its monodromy is the identity, so this boundary map is zero. The fundamental group action in the general construction is essential here.^[ref]