Attaching-belt intersection matrix of adjacent-index handles

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $W$ be a compact smooth $n$-manifold with collared boundary, let $1\le k\le n-2$, and let an index-ordered presentation attach $k$-handles $e_1,\dots ,e_m$ to a connected outgoing boundary $M$, then attach $(k+1)$-handles $g_1,\dots ,g_{\mathrm{\ell }}$ to the middle boundary $N={\mathrm{\partial }}_{+}(W\cup e_1\cup \cdots \cup e_m)$. Suppose every attaching sphere $A_i=g_i(S^k\times \{0\})$ meets every belt sphere $B_j$ of $e_j$ transversely. If the middle stage is oriented (the attachments extend the orientation of $W$) and the spheres carry the orientations induced by the handle framings and the induced boundary orientation of $N$, put $M_{ij}=I(A_i,B_j)\in \mathbb{Z}$, the oriented intersection number. Without orientations put $M_{ij}\in {\mathbb{Z}}_2$, the mod-2 intersection number. The resulting $\mathrm{\ell }\times m$ matrix is the attaching-belt intersection matrix of the adjacent-index handles. For a nontransverse configuration use the homotopy-invariant extensions in the cited intersection-number definitions. Whenever an isotopy of the attaching embeddings makes the configuration transverse, its entries are those same numbers; no separate existence theorem for an arbitrarily small embedding isotopy is asserted here. The computations on this page use configurations already transverse.

The matrix depends on the chosen index-ordered presentation, on the framings and orientations of the handles, but not on an isotopy used only to make the attaching spheres transverse to the fixed belt spheres (The oriented intersection number is homotopy invariant, The mod 2 intersection number is homotopy invariant). For $k=n-1$ the roles of the two families degenerate and the matrix is not defined here: the definition is restricted to the middle range $1\le k\le n-2$, and the endpoint indices $k=0$ and $k=n-1$ are treated separately. When every pair is transverse the entries are finite sums of local signs by Compact transverse complementary intersections are finite, and the fixed-transverse entries are choice-free; Countable Choice is inherited only from the published intersection-number definitions cited above.