The handle chain complex computes singular homology

For nonempty $M$, first rescale $f$ into $(0,1)$ by an increasing affine map; with both faces empty it is adapted. Apply the local separation of [F1], keeping the critical points and indices, construct the adapted field as in [F2], then apply the correspondence and index rearrangement of [F2]. The rearrangement puts indices in order. Equal-index consecutive handles can be made simultaneous by applying the separation and equal-value interchange arguments underlying [F2] within their index block; the crossing-sphere dimension inequality is $(k-1)+(n-k-1)<n-1$. Thus take stages $W_{-1}=\mathrm{\varnothing }\subseteq W_0\subseteq \cdots \subseteq W_n=M$, where $W_k$ adds the $m_k$ handles of index $k$ along disjoint attaching regions. For empty $M$ take every stage empty. In all cases set $W_j=\mathrm{\varnothing }$ for $j<0$ and $W_j=M$ for $j>n$, so all endpoint triples are defined.

By [F3], applied at the single level $W_k$, the relative group $H_j(W_k,W_{k-1};F)$ is zero for $j\mathrm{\ne }k$ and is an $F$-vector space of dimension $m_k$ for $j=k$; choose an orientation of each core disk, and take the resulting relative classes as its basis. Setting $C_k:=H_k(W_k,W_{k-1};F)$ (with $W_{-1}=\mathrm{\varnothing }$) gives graded $F$-vector spaces with ${\dim }_F{C}_k=m_k$, a basis as in (i); in particular $C_k=0$ for $k\mathrm{\notin }\{0,\dots ,n\}$.

Define ${\mathrm{\partial }}_k:C_k\to C_{k-1}$ as the composite of the connecting map ${\delta }_k:C_k\to H_{k-1}(W_{k-1};F)$ of the pair $(W_k,W_{k-1})$ with the quotient map $H_{k-1}(W_{k-1};F)\to H_{k-1}(W_{k-1},W_{k-2};F)=C_{k-1}$. By the factorization clause of [F5] this is exactly the boundary homomorphism of the triple $W_k\supseteq W_{k-1}\supseteq W_{k-2}$, which is assertion (ii). It is a homomorphism of $F$-vector spaces, and for $k=0$ the target is zero.

Auxiliary computation: $H_j(W_k;F)=0$ whenever $j>k$. For $k=0$ this is immediate since $W_0$ is a disjoint union of disks. For the induction step and $j>k$, step 2.1 makes both relative terms vanish in the exact sequence $0=H_{j+1}(W_k,W_{k-1})\to H_j(W_{k-1})\to H_j(W_k)\to H_j(W_k,W_{k-1})=0$. Thus $H_j(W_k)\cong H_j(W_{k-1})=0$.

${\mathrm{\partial }}^2=0$: the composite ${\mathrm{\partial }}_{k-1}{\mathrm{\partial }}_k$ is the composite $C_k\stackrel{{\delta }_k}{\to }{H}_{k-1}(W_{k-1})\stackrel{q_{k-1}}{\to }{C}_{k-1}\stackrel{{\delta }_{k-1}}{\to }{H}_{k-2}(W_{k-2})\stackrel{q_{k-2}}{\to }{C}_{k-2},$ and the middle two arrows compose to zero by exactness of the pair sequence of $(W_{k-1},W_{k-2})$ at the node $H_{k-1}(W_{k-1},W_{k-2})$ (the image of the quotient map is the kernel of the connecting map); hence ${\mathrm{\partial }}_{k-1}{\mathrm{\partial }}_k=0$, which is (iii).

By step 3.2, $H_k(W_{k-1};F)=0$ and $H_{k-1}(W_{k-2};F)=0$. The pair sequences therefore show that $i_k:H_k(W_k;F)\to C_k$ is injective with image $\ker {\delta }_k$, and that $q_{k-1}:H_{k-1}(W_{k-1};F)\to C_{k-1}$ is injective. Since ${\mathrm{\partial }}_k=q_{k-1}{\delta }_k$, it follows that $\ker {\mathrm{\partial }}_k=\ker {\delta }_k=i_k(H_k(W_k;F))$. For $k=0$ the target is zero and the same conclusion follows from $W_{-1}=\mathrm{\varnothing }$.

From the pair sequence of $(W_{k+1},W_k)$, whose relative group vanishes in degree $k$ by step 2.1, there is an exact tail $C_{k+1}\stackrel{{\delta }_{k+1}}{\to }{H}_k(W_k;F)⟶H_k(W_{k+1};F)⟶0,$ so $H_k(W_{k+1};F)\cong H_k(W_k;F)\mathrm{/}\mathrm{im}{\delta }_{k+1}$; under the injection $i_k$ of step 4.2 the subspace $\mathrm{im}{\delta }_{k+1}$ corresponds exactly to $\mathrm{im}{\mathrm{\partial }}_{k+1}=\mathrm{im}(q_k{\delta }_{k+1})$. Taking quotients gives $H_k(C_{\bullet })=\ker {\mathrm{\partial }}_k\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{k+1}\cong H_k(W_k;F)\mathrm{/}\mathrm{im}{\delta }_{k+1}\cong H_k(W_{k+1};F)\mathrm{.}$

Finally $H_k(W_{k+1};F)\cong H_k(M;F)$: the remaining handles, attached to $W_{k+1}$, all have index at least $k+2$, so [F4] with $q=k+2$ gives an isomorphism in degree $k$; when $k=n$ no handles remain and $W_{n+1}=W_n=M$. Combining with step 5.1 gives $H_k(C_{\bullet })\cong H_k(M;F)$, which is (iv). Since ${\dim }_F{C}_k=m_k<\mathrm{\infty }$ by step 2.1, every $H_k(C_{\bullet })$ is finite-dimensional and vanishes for $k>n$, and so does $H_k(M;F)$. This transposes the proof that cellular homology computes singular homology from the CW filtration to the handle filtration, using the concentration of step 2.1 in place of the skeletal concentration.