Homology lemma: a handle-basis class is realized by a sphere meeting the belt sphere once

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $W$ be a compact smooth $(n+1)$-manifold with a finite handle decomposition relative to ${\mathrm{\partial }}_{0}W$ in which all handles have index $\ge q$, where $2\le q\le n-3$, and suppose the outgoing boundary ${\mathrm{\partial }}_1{W}_q$ is simply connected (Simply connected topological spaces). Let $f:S^q\hookrightarrow {\mathrm{\partial }}_1{W}_q$ be an embedded sphere whose class in $C_q=H_q(W_q,W_{q-1};\mathbb{Z})$ equals $\pm [\phi ]$ for the basis element $[\phi ]$ of some $q$-handle $\phi$ (The relative handle chain complex computes $H_{\ast }(W,M_0)$ and has the intersection matrix as its differential). Then $f$ is isotopic in ${\mathrm{\partial }}_1{W}_q$ to an embedding that meets the belt sphere of $\phi$ transversely in exactly one point and is disjoint from the belt spheres of all other $q$-handles (K handle core cocore attaching region and belt sphere). In the special case that the presentation consists of two adjacent index classes $q,q+1$, the hypothesis on the class says exactly that the intersection vector of $f$ with the belt spheres is $\pm$ a standard basis vector.

For $q=2$, also assume that ${\pi }_1({\mathrm{\partial }}_{0}W)\to {\pi }_1(W_2)$ is injective. In the h-cobordism applications this follows from the incoming homotopy equivalence and the absence of later handles below index three; nullhomotopic $2$-handle attaching circles also suffice.

The wider arbitrary-sphere endpoint $q=n-2$ from the source is not proved by this lemma: it would require an additional argument controlling the complement of the arbitrary codimension-two sphere. Actual two-index h-cobordism attaching spheres admit that control by reversing the handles, and the geometric middle-cancellation lemma proves that endpoint directly.