The based handle chain complex over the fundamental group ring

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $(W;M_0,M_1)$ be a nonempty connected compact smooth cobordism triad with a finite index-ordered handle presentation $H$ relative to $M_0$ (Smooth cobordism triad for Morse theory, Handle decomposition relative to the incoming boundary). Fix a finite CW model $K\simeq M_0$ when $M_0$ is nonempty; such models exist by A handle decomposition gives a relative CW complex. If $M_0$ is empty, use $K=\mathrm{\varnothing }$. Transfer the handle attaching maps to this model using a chosen homotopy inverse and the cell-attachment induction of A handle decomposition gives a relative CW complex. Cellular approximation at each finite stage gives a finite CW pair $(X,K)\simeq (W,M_0)$ with one relative cell per handle (Cellular approximation for maps of CW pairs). Fix this model and its comparisons throughout. In subsequent notation for cellular chains, $W,M_0$ mean these chosen models $X,K$.

Put $\pi ={\pi }_1(W)$ and $R=\mathbb{Z}[\pi ]$ (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements). If the incoming inclusion induces a fundamental-group isomorphism, identify $\pi$ with ${\pi }_1(M_0)$ along that inclusion, as in every h-cobordism application. Choose a universal cover $p:\tilde{X}\to X$; $p^{-1}K$ is its induced cover of $K$, and need not be a universal cover unless the incoming fundamental-group map is an isomorphism (Universal covering spaces, Based cellular chains of a universal cover as finite free right group-ring modules).

The based handle chain complex is $C_k^{\mathrm{h}}(W,M_0;H):=C_k^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(\tilde{X},p^{-1}K;R),$ where the semicolon records the deck-induced module structure and the homology coefficients defining cellular chains are integral. Its right action is $c\cdot g=T_{g^{-1}}c$ (Right action on universal-cover chains, For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group). An orientation of each handle core and one oriented lift of its relative cell give a basis $[h]$ in the handle's degree. The complex is bounded finite based free over $R$; an empty handle list gives the zero complex.

In these bases the relative cellular boundary is $d[h]={\sum }_{h^{\mathrm{\prime }}}[h^{\mathrm{\prime }}]\cdot a_{h^{\mathrm{\prime }}h},a_{h^{\mathrm{\prime }}h}={\sum }_{g\in \pi }{\lambda }_{h^{\mathrm{\prime }},g}g,$ with $h^{\mathrm{\prime }}$ ranging over handles one index lower. For transverse middle-level data, ${\lambda }_{h^{\mathrm{\prime }},g}$ is the incidence count of the chosen lifted attaching sphere against the belt of the lifted handle $T_{g^{-1}}{\tilde{h}}^{\mathrm{\prime }}$, with its transported core and normal orientations. For a lifted lower $k$-handle, the relevant collapse on its outgoing region $D^k\times S^{\dim W-k-1}$ is $(x,y)\mapsto [x]\in D^k\mathrm{/}{S}^{k-1}$, with the attaching rim and all other lower pieces sent to the quotient basepoint. The fibre over the interior point $[0]$ is exactly its belt sphere. Transversality makes $[0]$ a regular value of the restricted upper attaching sphere, and each local degree is its attaching-belt sign. Summing these local degrees in every lift gives the formula (the incidence identity of Handle boundary coefficients are attaching-belt intersection numbers); only finitely many lifted belts meet the compact attaching sphere. This defines the labels unambiguously in the right convention: the matrix has lower handles as rows and upper handles as columns, and coordinate columns are multiplied by that matrix on the left. If $W$ is oriented, these incidence signs are the ordinary attaching-belt signs in each oriented lift; without global orientability, use the lifted local core/normal orientations rather than a nonexistent global boundary orientation. Reduction modulo two forgets the signs.

With the CW model fixed, reordering, reorienting or relifting handle generators changes the bases by permutations and units $\pm g$. This definition does not assert independence of the chosen finite CW model or compare arbitrary presentations.