PTFA localized boundary intersection rank comparison

Apply [F1] to the actual $k[G]$ singular complex of $W$ and to each of the finitely many boundary components. Tensoring the specified chain equivalences and homotopies with $k$ or $K$ preserves their equations. On singular generators augmentation sends every group transport to $1$, so these are exactly the ordinary $k$ and twisted $K$ chains. The bounded projective models show that all homology dimensions are finite. By [F2] their alternating homology dimensions are equal; denote the common number by $\chi$. Apply [F1] over $k$ as well when needed. If a coefficient homomorphism is trivial, its actual singular $K$ complex is the constant singular $k$ complex tensored with $K$. Its bounded projective $k$ model is a bounded finite-dimensional vector-space complex; finite elimination splits it into its homology and contractible pairs. Tensoring those finite splittings with $K$ proves equality of all ordinary and $K$ Betti numbers in the trivial case, without a choice of a basis in an infinite vector space.

For a connected space with nontrivial $\varphi$, $H_0(-;K)=0$: paths identify all point fibers with the base fiber in degree zero, and a based loop with image $g\mathrm{\ne }1$ imposes $(g-1)K=K$, because the nonzero element $g-1$ is invertible in $K$. This argument uses one path for each individual zero-cycle, not a simultaneous path selection. If the homomorphism is trivial, $H_0(-;K)=K$. For $W$ write ${\beta }_i={\dim }_k{H}_i(W;k)$ and $b_i={\dim }_K{H}_i(W;K)$. If its homomorphism is nontrivial, ${\beta }_0=1,b_0=0$, so [F2] gives $b_1\le {\beta }_1-1$. This uses the nonnegative bounded projective model from step 1.1, not an unproved finite CW structure.

Here is the exact rank formula for either coefficient system. Write $c_i$ for its dimensions on $W$ and $a_i$ for those on $M$. Finite projective models over the division field identify dual cohomology dimensions with homology dimensions by finite elimination, and [F3] therefore gives $H_i(W,M)$ dimension $c_{4-i}$ and boundary dimensions $a_i=a_{3-i}$. If $M$ is nonempty, $H_4(W)=0$: under relative duality it is degree-zero relative cohomology, whose parallel sections vanishing on the nonempty boundary vanish on connected $W$. The top of the pair sequence is then $0\to H_4(W,M)\to H_3(M)\to H_3(W)\to H_3(W,M)\to H_2(M)\to H_2(W)\mathrm{.}$ Successive image dimensions are $c_0$, $a_0-c_0$, $c_3-a_0+c_0$, and $c_1-c_3+a_0-c_0$. Hence the image of $H_2(M)\to H_2(W)$ has dimension $a_1-c_1+c_3-a_0+c_0$, and its cokernel has dimension $r_{\ast }=c_2-a_1+c_1-c_3+a_0-c_0=\chi +2c_1-2c_0-a_1+a_0\mathrm{.}$ In the last equality $H_4(W)=0$ and higher homology vanishes by duality. If $M$ is empty, closed duality gives $c_4=c_0,c_3=c_1$ and $r_{\ast }=c_2$, yielding the same formula with $a_0=a_1=0$. Thus the formula is valid in both boundary cases.

For a boundary component write ${\beta }_i^j,b_i^j$ similarly. If its restricted homomorphism is trivial, step 1.1 gives $b_i^j={\beta }_i^j$. Otherwise step 2.1 and [F2], applied to that component's projective model, give $b_0^j=0$ and $b_1^j\le {\beta }_1^j-1=0$. Thus $b_1^j=0$, while ${\beta }_0^j={\beta }_1^j=1$. In both cases ${\beta }_1^j-b_1^j={\beta }_0^j-b_0^j\mathrm{.}$ Summing over the finitely many components gives the identical equality for the total boundary Betti numbers ${\beta }_i(M),b_i(M)$. It also holds for empty boundary, with all four numbers zero.

If $\varphi$ on $W$ is trivial, equality of all Betti numbers and natural ordinary coefficient base change in step 1.1 give equality of the two cokernel dimensions, or equivalently the identical formulas of step 2.2 do. Suppose it is nontrivial. Subtract the $K$ formula from the $k$ formula, using the common Euler number and ${\beta }_0=1,b_0=0$: $r_k-r_K=2({\beta }_1-1-b_1)-({\beta }_1(M)-b_1(M))+({\beta }_0(M)-b_0(M))\mathrm{.}$ The last two terms cancel by step 3.1, and the first is nonnegative by step 2.1. This proves the claimed inequality for empty or nonempty boundary. Every chain-model construction used actual domination maps and a finite projective strictification; every subsequent elimination and exact-sequence dimension computation was finite. Therefore no choice principle or smooth/PL/CW assumption on the manifold has been introduced.