The middle-handle intersection matrix of an h-cobordism

In the situation of the concentration lemma (Trading concentrates a simply connected h-cobordism in two adjacent middle indices), choose an orientation of $W$. Such an orientation exists: choose one at an interior basepoint and transport it along paths by Orientation local system and orientation cover. Transport depends only on endpoint-fixed path homotopy, so simple connectivity makes its value at each point unique. In each chart the transported value is constant in the orientation-cover coordinate; hence it is a continuous local orientation. The boundary collars extend this orientation to $W$ and induce one on each level. This is a single initial choice, not a family of independent choices.

Let the presentation of the h-cobordism have handles $e_1,\dots ,e_r$ of index $k$ and $g_1,\dots ,g_r$ of index $k+1$, $2\le k\le n-2$, attached at the two consecutive levels, and let $N$ be the outgoing boundary after the $k$-handles (K handle core cocore attaching region and belt sphere). After an arbitrarily small isotopy of the attaching data the attaching spheres $A_i$ of the $g_i$ and the belt spheres $B_j$ of the $e_j$ are transverse, and the middle-handle intersection matrix of the h-cobordism for the chosen presentation is

$M=(I(A_i,B_j))\in M_r(\mathbb{Z}),$

the matrix of oriented intersection numbers in $N$ (The oriented intersection number, The local oriented intersection sign). The sphere orientations use the core and belt-normal convention of the relative handle-chain proposition and the boundary orientation of $N$ when $W$ is oriented (Oriented smooth manifolds and oriented charts); without orientations the matrix $({\mathrm{\#}}_2(A_i\cap B_j))\in M_r(\mathbb{Z}\mathrm{/}2)$ with entries the mod-two intersection numbers is defined (The mod 2 intersection number).

By the relative handle chain complex (The relative handle chain complex computes $H_{\ast }(W,M_0)$ and has the intersection matrix as its differential) this matrix represents ${\mathrm{\partial }}_{k+1}:C_{k+1}\to C_k$ on row coordinate vectors, $x\mapsto xM$; the matrix on column coordinate vectors is $M^T$. For a fixed handle count, slides, sign changes and renumbering give the corresponding basis changes of this differential. Presentation changes may also insert or delete an isolated geometrically cancelling $k\mathrm{/}(k+1)$ pair (Creation of a cancelling handle pair, Handle cancellation). Its attaching and belt spheres have no incidences with the old handles and meet one another once, so the new matrix is $M\oplus (\epsilon )$, $\epsilon =\pm 1$, or $M\oplus (1)$ after reorientation. In particular the empty and one-pair presentations of $S^n\times [0,1]$ have matrices of sizes $0$ and $1$; presentation-dependence includes stabilization, which cannot be produced by size-preserving basis changes alone. The matrix is only readable for a configuration that is already transverse; if some pair is not transverse, the matrix is read after a small isotopy, which does not change the presented manifold. Countable choice is inherited from the intersection-number and handle suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

For fixed oriented attaching data the entries are independent of the small transverse isotopy, by the homotopy-invariance clause of The oriented intersection number; the same holds modulo two by The mod 2 intersection number. For $r=0$ the matrix is empty and the corresponding differential is the unique isomorphism $0\to 0$.

The transverse isotopy can be chosen arbitrarily small: on the compact middle level $N$, finitely many chart vector fields multiplied by compactly supported bumps span $TN$ (A manifold bump for a compact set inside an open set). Compose their small-time flows to obtain a finite-parameter family $H_s$ of diffeomorphisms with $H_0=\mathrm{i}\mathrm{d}$ (Compactly supported smooth vector fields are complete, The fundamental theorem on flows). The evaluation $(x,s)\mapsto H_s(x)$ is a submersion for $s$ in a sufficiently small parameter ball, by the spanning condition at $s=0$ and compactness. Applied to the disjoint union of attaching spheres, parametric transversality to each of the finitely many fixed belt spheres gives parameters arbitrarily near zero that are good for all pairs (Parametric transversality). The isotopy $t\mapsto H_{ts}$ applied to all upper attaching embeddings transports their framings and preserves their mutual disjointness; the belt spheres remain the fixed comparison family.