Handle decomposition

A handle decomposition builds an $n$-dimensional manifold by successively adjoining $D^h\times D^{n-h}$ along $S^{h-1}\times D^{n-h}$. The attaching data comprise an attaching sphere and its framing of an embedded sphere. Reading a decomposition backwards replaces index $h$ by index $n-h$ and interchanges its attaching spheres and belt spheres.

In a critical level handle embedding, the handles occur at distinct heights in an ambient product with the real line. Between consecutive heights, the boundary of the already attached handles travels through a product collar neighbourhood. At a handle height the horizontal slice contains the handle, attached to the incoming boundary along its attaching region. This separates the local topology changes from the intervening product regions.

Write each sphere as two disks, $S^2=D_{-}^2\cup D_{+}^2$. In their product, $D_{-}^2\times D_{-}^2$ is a zero-handle; $D_{+}^2\times D_{-}^2$ and $D_{-}^2\times D_{+}^2$ are two-handles. Their attaching circles are the core circles of the complementary solid tori in $\partial (D^2\times D^2)$, hence a Hopf link. Constant normal directions from the disk factors give both components the zero Seifert framing. The remaining $D_{+}^2\times D_{+}^2$ is a four-handle. This product construction establishes the actual diffeomorphism type, rather than relying on its intersection form alone. The resulting intersection matrix is $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.

Attach two one-handles to a four-dimensional zero-handle, then a two-handle along the commutator of their generators with the product zero framing. The result is $T^2\times D^2$. Its spine is the usual cell structure of the torus: one vertex, two one-cells, and a two-cell attached by $xy{x}^{-1}{y}^{-1}$. An embedded representative of its $H_2$ generator is obtained by taking a once-punctured torus carried by the two one-handles and capping its boundary by the core disk of the two-handle. It meets the relative co-core once and has self-intersection zero. The boundary is $T^3$, whose integral homology has ranks $1,3,3,1$.

A handle slide replaces the framed attaching sphere of one handle by a band sum with a parallel copy of another handle of the same index. It changes the attaching data while preserving the diffeomorphism type of the resulting manifold. On the handle incidence matrix it performs an elementary row operation, or a column operation when applied to the dual decomposition.

In the intermediate-index range, if an attaching sphere $A_1$ meets a belt sphere $B_1$ at exactly one transverse point, handle slides can remove the intersections of every other attaching sphere with $B_1$. The dual slides then remove the intersections of $A_1$ with the other belt spheres. Each move uses a band to the unique pivot sheet and a local isotopy removing the old and new intersections. If $m_{ji}=A_j\cdot B_i$ and $\delta =m_{11}=\pm 1$, the remaining incidence block is $m_{ji}-m_{j1}{m}_{1i}/\delta$, the Schur complement of the pivot. Extreme-index cases need separate hypotheses.

The belt sphere of an index-$h$ handle is $\{0\}\times S^{n-h-1}$ in the boundary after attachment. It is the attaching sphere of the dual index-$(n-h)$ handle. Its intersections with the attaching spheres of later handles detect passages through this handle.

The attaching sphere of an index-$h$ handle $D^h\times D^{n-h}$ is $S^{h-1}\times \{0\}$ in the previous boundary. Its framing of an embedded sphere identifies a neighbourhood with $S^{h-1}\times D^{n-h}$ and specifies the handle attachment.