A handle slide realizes an elementary row operation

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].