Handle attachments are relative cell attachments up to homotopy

Attaching a smooth handle with corner rounding: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $X$ be a smooth $n$-manifold with boundary, and let $k$ be an integer with $0\le k\le n$. Attach the handle of K handle core cocore attaching region and belt sphere by a smooth embedding $h:S^{k-1}\times D^{n-k}\to \mathrm{\partial }X$ that extends to a neighborhood of the disk factor. Form the quotient of $X\bigsqcup (D^k\times D^{n-k})$ identifying $z$ with $h(z)$ in the attaching region. The disk coordinates trivialize the normal bundle of the attaching sphere; this framing is part of the data. Use collars from Collar neighborhood theorem to give the seam its product smooth charts, then round the compact codimension-two corner. There is no corner to round when $k=0$ or $k=n$.

K handle core cocore attaching region and belt sphere: For integers $0\le k\le n$, the standard $n$-dimensional $k$-handle is $D^k\times D^{n-k}$. Its core is $D^k\times \{0\}$, its cocore is $\{0\}\times D^{n-k}$, its attaching region is $S^{k-1}\times D^{n-k}$, and its attaching sphere is $S^{k-1}\times \{0\}$. The outgoing region is $D^k\times S^{n-k-1}$ and the belt sphere is $\{0\}\times S^{n-k-1}$. Here $D^j$ is the closed unit disk, $D^0$ is a point, and $S^{-1}=\mathrm{\varnothing }$.

Cell attachment by a characteristic map: For a space $X$, an attaching map $f:S^{n-1}\to X$, and $n\ge 1$, attach an $n$-cell by the pushout $X{\cup }_f{D}^n=(X\bigsqcup D^n)\mathrm{/}(z\sim f(z)$ for $z\in S^{n-1})$. The quotient map restricted to $D^n$ is its characteristic map; its image is the closed cell and the image of ${\mathring{D}}^n$ is the open cell. For $n=0$, use $S^{-1}=\mathrm{\varnothing }$, so $X{\cup }_f{D}^0=X\bigsqcup \{\ast \}$.

Cofibration and homotopy extension property: A continuous map $i:A\to X$ has the homotopy extension property (HEP), or is an unbased cofibration, if for every target $Z$, continuous $f:X\to Z$, and continuous $h:A\times I\to Z$ satisfying $h(a,0)=f(i(a))$, there is a continuous $H:X\times I\to Z$ with $H(x,0)=f(x)$ and $H(i(a),t)=h(a,t)$. No uniqueness is required.

Handle retraction. There is a continuous homotopy $H_t:D^k\times D^{n-k}\to D^k\times D^{n-k}$, $t\in [0,1]$, with $H_0=\mathrm{i}\mathrm{d}$, $H_t$ the identity on the attaching region $S^{k-1}\times D^{n-k}$ for every $t$, and $H_1$ mapping onto $(S^{k-1}\times D^{n-k})\cup (D^k\times \{0\})$. Explicitly, with $r=\mathrm{\mid }u\mathrm{\mid }$, let $\eta (r):=\min (2r,1)$ and $\chi$ continuous with $\chi (r)=0$ for $r\le 1\mathrm{/}2$, $\chi (r)=1$ for $r\ge 3\mathrm{/}4$ and $0\le \chi \le 1$, and put $H_t(u,v):=(u{\nu }_t(r)\mathrm{/}r,v{\psi }_t(r))$ with ${\nu }_t(r):=(1-t)r+t\eta (r)$ and ${\psi }_t(r):=(1-t)+t\chi (r)$, the ratio at $u=0$ read as ${\nu }_t(r)\mathrm{/}r\le 2$. Then $u{\nu }_t(r)\mathrm{/}r$ has norm $(1-t)r+t\eta (r)\le 1$, so $H_t$ takes values in the handle and is continuous; for $r=1$ one has $\eta (1)=1=\chi (1)$, so $H_t(u,v)=(u,v)$ and $H_t$ is the identity on the attaching region. At $t=1$ the first coordinate $u\eta (r)\mathrm{/}r$ has norm $\eta (r)$, which equals $2r\in [0,1]$ for $r\le 1\mathrm{/}2$ and $1$ for $r\ge 1\mathrm{/}2$, and the second coordinate $v\chi (r)$ vanishes for $r\le 1\mathrm{/}2$; hence the image lies in the union, the values with $r\le 1\mathrm{/}2$ cover $D^k\times \{0\}$ and the values with $r\ge 3\mathrm{/}4$ cover $S^{k-1}\times D^{n-k}$. This is Wall's handle retraction.