Handle Cancellation Slides and Elementary Moves — Examples

Given: Dimension $n\ge 1$, a compact manifold $W$, a $0$-handle $h^0$ attached as a disjoint ball $D^n$ and a $1$-handle $h^1$ attached by an embedding $S^0\times D^{n-1}\to {\mathrm{\partial }}_{+}(W\bigsqcup D^n)$ whose attaching $0$-sphere has one point on the new boundary sphere $S^{n-1}$ of $D^n$ and one point on ${\mathrm{\partial }}_{+}W$.

Geometrically cancelling adjacent handle pair and K handle core cocore attaching region and belt sphere: the belt sphere of a $0$-handle is the new boundary sphere $S^{n-1}$ (disconnected when $n=1$), the attaching sphere of a $1$-handle is a $0$-sphere, and the endpoint convention counts exactly one point of the $0$-sphere lying in the belt sphere, transversality being automatic.

The standard complementary pair fills a ball: for the standard embedding the union of the standard $0$-handle and the standard $1$-handle is an $n$-disc: in its case $k=0$ the standard complementary pair fills the ball, i.e. $(S^0\times D^n)\cup h^1\cong D^n$ for the standard hemisphere embedding.

Handle cancellation, Attaching a smooth handle with corner rounding and The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): assume ${\mathrm{A}\mathrm{C}}_{\omega }$; a geometrically cancelling pair may be deleted, giving a diffeomorphism relative to the incoming boundary; attachments are formed with corners rounded. The assumption is used through the standard-pair model of [F2] and through this deletion.

Given: Dimension $n=2$: a closed disc $D^2$ as a $0$-handle, a $1$-handle (band) attached along two disjoint intervals of $\mathrm{\partial }{D}^2$, with attaching orientations chosen so the disc orientation extends over the band, and then a $2$-handle (disc) attached along one boundary circle of the resulting annulus.

Attaching a smooth handle with corner rounding and K handle core cocore attaching region and belt sphere: the $1$-handle $D^1\times D^1$ has attaching region $S^0\times D^1$ and outgoing region $D^1\times S^0$, so its belt sphere in the outgoing boundary is a $0$-sphere $\{0\}\times S^0$; the $2$-handle $D^2\times D^0$ has attaching sphere $S^1\times \{0\}$.

Geometrically cancelling adjacent handle pair: in dimension $n=2$ and $k=1$ the attaching sphere of the $2$-handle is a circle and the belt sphere of the $1$-handle is a $0$-sphere in the middle boundary (the two boundary circles of the annulus); they meet transversely in isolated points, and the pair is geometrically cancelling when exactly one point of the belt sphere lies on the attaching circle.

Handle cancellation, The local oriented intersection sign, The mod 2 intersection number and The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): assume ${\mathrm{A}\mathrm{C}}_{\omega }$; a geometrically cancelling pair may be deleted, so the total is diffeomorphic to the initial disc; the unique transverse intersection has local sign $\pm 1$ and mod-2 count $1$.

Given: Dimension $n=4$; the $0$-handle $D^4$ with the standard $1$-handle $h^1$ attached, middle boundary $N={\mathrm{\partial }}_{+}(D^4\cup h^1)$ and belt sphere $B\subseteq N$ of $h^1$; two embedded circles ${\gamma }_1,{\gamma }_2\subseteq N$ with disjoint images, with ${\gamma }_1$ meeting $B$ transversely in exactly one point and ${\gamma }_2\cap B=\mathrm{\varnothing }$; the $2$-handles $g_1,g_2$ attached to $N$ along ${\gamma }_1,{\gamma }_2$ with chosen framings.

Attaching a smooth handle with corner rounding and K handle core cocore attaching region and belt sphere: in dimension $4$ a $1$-handle has attaching region $S^0\times D^3$, outgoing region $D^1\times S^2$ and belt sphere $S^2$; a $2$-handle has attaching sphere $S^1$ and attaching region $S^1\times D^2$.

The standard complementary pair fills a ball: attaching the standard $1$-handle to $D^4$ along the equatorial embedding gives $D^4\cup h^1\cong S^1\times D^3$ with outgoing boundary $S^1\times S^2$, and up to this diffeomorphism the belt sphere of $h^1$ is the fiber sphere $\{p\}\times S^2$.

Attaching-belt intersection matrix of adjacent-index handles: for $n=4$ and $k=1$ the matrix is defined ($1\le k\le n-2=2$), its rows are the $(k+1)$-handles $g_i$ and its columns the $k$-handles $e_j$, and $M_{ij}$ is the intersection number of the attaching sphere of $g_i$ with the belt sphere of $e_j$ in the middle boundary, over $\mathbb{Z}$ or ${\mathbb{Z}}_2$ according to the orientations.

The oriented intersection number, The local oriented intersection sign and The mod 2 intersection number: transverse complementary-dimensional intersections are finite with local signs $\pm 1$; a single transverse point of a circle with a $2$-sphere has entry $\pm 1$ over $\mathbb{Z}$ and $1$ over ${\mathbb{Z}}_2$, and disjoint spheres have entry $0$.

Handle slide of one k handle over another and Handle slides preserve the relative diffeomorphism type: assume ${\mathrm{A}\mathrm{C}}_{\omega }$; a slide of $g_2$ over $g_1$ is defined here (index $2$, middle boundary of dimension $3$, so $1\le 2\le 3-1$), replaces the attaching circle of $g_2$ by the band sum with a framed parallel copy of the attaching circle of $g_1$, and leaves the total $4$-manifold unchanged.

Handle slides act by elementary basis change on handle chains and Elementary matrix operations are realized by handle slides: assume ${\mathrm{A}\mathrm{C}}_{\omega }$; under the disk-push diffeomorphism together with its specified lower-stage homotopy, the slid core satisfies $[C_2^{\mathrm{\prime }}]=[C_2]\pm [C_1]$ in the handle chain group, so intersecting the fixed belt sphere with both sides gives $M_{2,1}^{\mathrm{\prime }}=M_{2,1}\pm M_{1,1}$, the elementary row operation $R_2\mapsto R_2\pm R_1$; this is case (ii) of the matrix-operation proposition because $k+1=2\le n-2=2$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): ${\mathrm{A}\mathrm{C}}_{\omega }$ is assumed; it is used through [F5] and [F6].

Given: The standard $4$-dimensional model $W=D^4\cup h^2$ with middle boundary $N\cong S^2\times S^1$, belt sphere $B=\{p\}\times S^1$, and a $2$-sphere $A\subseteq N$ obtained from $S^2\times \{u_0\}$ by a finger move across $B$.

Attaching-belt intersection matrix of adjacent-index handles and K handle core cocore attaching region and belt sphere: the belt sphere of the $2$-handle in the middle boundary is a circle, and the attaching sphere of a $3$-handle attached to $N$ is a $2$-sphere; the matrix entry is the oriented, respectively mod-2, intersection number of the attaching sphere with the belt sphere.

Algebraic cancellation does not yet give geometric cancellation: on $N=S^2\times S^1$ there are an embedded $2$-sphere and an embedded circle meeting transversely in exactly three points with local signs $+1,+1,-1$, so that the oriented intersection number is $1$ although the geometric intersection has three points; the configuration is realized with the sphere as the attaching sphere of a $3$-handle and the circle as the belt sphere of the $2$-handle in the standard model.

The oriented intersection number, The local oriented intersection sign and The mod 2 intersection number: the oriented number is the sum of local signs over the transverse intersection, and the mod-2 number is its cardinality modulo two.

Geometric cancellation is a unit entry in the handle matrix and Handle cancellation: a single transverse point gives a unit entry, and only then does the cancellation theorem apply; a unit entry does not by itself supply a single geometric intersection point.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): ${\mathrm{A}\mathrm{C}}_{\omega }$ is assumed; it is used through the intersection-number and cancellation suppliers.

Given: In dimension $n=3$ the manifold $W_0=D^3$ with a $1$-handle $h^1$ attached along two disks of $\mathrm{\partial }{D}^3$, giving $W_1=D^3\cup h^1\cong S^1\times D^2$, and a $2$-handle $h^2$ attached along an embedded circle $\gamma \subseteq \mathrm{\partial }{W}_1$ that bounds a closed disk in $\mathrm{\partial }{W}_1$ disjoint from the belt sphere $S^1$ of $h^1$.

Attaching a smooth handle with corner rounding and K handle core cocore attaching region and belt sphere: attaching a $1$-handle to $D^3$ along two disks of the boundary and rounding the corner gives the solid torus $S^1\times D^2$, whose boundary is a torus; the belt sphere of the $1$-handle is the meridian $\{p\}\times S^1$ of that torus, a $2$-handle attaches along a circle, and a circle in the boundary that bounds a disk there is null-homotopic in the solid torus.

Attaching-belt intersection matrix of adjacent-index handles and Geometric cancellation is a unit entry in the handle matrix: the matrix entry is the intersection number of $\gamma$ with the belt sphere; if the two are disjoint the entry is $0$ over both $\mathbb{Z}$ and ${\mathbb{Z}}_2$, whereas a geometrically cancelling pair would have a unit entry.

Seifert–van Kampen identifies the fundamental group with a group pushout: for two open path-connected sets with path-connected overlap, the fundamental group is the pushout of the two groups over the overlap group. A collar thickening of an attached $2$-handle gives such a cover with the old manifold and handle as deformation retracts, and overlap retracting onto the attaching annulus $S^1\times D^1$.

Handle cancellation and Geometrically cancelling adjacent handle pair: a cancelled pair may be deleted, so if $(h^1,h^2)$ cancelled then $W_1\cup h^2$ would be diffeomorphic to $D^3$ relative to the lower stage, in particular simply connected.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): ${\mathrm{A}\mathrm{C}}_{\omega }$ is assumed; it is used through [F3] and [F4].