A nonorientable closed manifold has no integral fundamental class

Apply the choice-free part of [F1] with $n=2$. It makes $P$ a nonempty compact connected boundaryless $2$-manifold and supplies a canonical mod-two fundamental class. In the quotient model a specified point is $x=[1:0:0]$. By [F4], its local integral top-homology group is infinite cyclic and therefore has nonzero generators. The same holds at every point.

The integral cellular chain complex of [F2] is $0⟶\mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\stackrel{0}{\to }\mathbb{Z}⟶0,$ with the three nonzero terms in degrees $2,1,0$. Multiplication by two is injective on $\mathbb{Z}$, so the degree-two cycle group is zero. There is no degree-three cell and hence no incoming boundary. Therefore [F3] gives $H_2(P;\mathbb{Z})=\ker (2)\mathrm{/}0=0.$ For clarity the same complex gives $H_1(P;\mathbb{Z})=\mathbb{Z}\mathrm{/}2$ and $H_0(P;\mathbb{Z})=\mathbb{Z}$; the failure is in the required top degree, not in connectedness.

Every homomorphism from the zero group $H_2(P;\mathbb{Z})$ has image zero. In particular restriction at the explicit point $x$ of step 1.1 cannot hit either generator of its infinite cyclic stalk. Thus no global integral class can have the required generator restrictions at all points. If $P$ had an integral orientation, [F5] with the hypotheses in step 1.1 would make that restriction onto, contradicting step 1.2. Hence $P$ is nonorientable and witnesses the failure of an ordinary integral fundamental class without the orientation hypothesis.

Modulo two, the differential $2$ in step 1.2 becomes zero, as established through characteristic coefficient maps in [F2]; the entire top cellular group survives as $H_2(P;{\mathbb{F}}_2)={\mathbb{F}}_2$. The canonical class from step 1.1 restricts to the nonzero mod-two local generator, so cannot be zero. It is therefore the unique nonzero element of this one-dimensional group. This is a change of coefficients, not an integral generator hidden by an orientation convention.

The example uses the positive even dimension two, not the zero-dimensional point ${\mathbb{R}\mathbb{P}}^0$, which is orientable. The zero integral top group in step 1.2 is compared with a nonzero stalk at a specified point, so the failure is not vacuous. The two endpoints of the one-cell attach to the sole zero-cell, giving $d_1=0$, while the top attaching incidence is $2$, not zero; both are covered by [F2]'s full attaching calculation. Degenerate singular chains are included in the comparison [F3]. No top cap isomorphism or field-cohomology dualization is used, and every clause of [F1] and [F2] invoked above is explicitly choice-free.