Elementary expansions and collapses of finite CW complexes

Let $X$ be a finite CW complex and let $n\ge 1$. Write $D^n=\{x\in {\mathbb{R}}^n:\mathrm{\mid }x\mathrm{\mid }\le 1\}$ for the closed unit ball, $S^{n-1}=\mathrm{\partial }{D}^n$ for its boundary sphere, and $D_{+}^{n-1}=\{(x,t)\in S^{n-1}:t\ge 0\}$ for a closed upper hemisphere of $S^{n-1}$.

An elementary expansion of $X$ of dimension $n$ is an inclusion $X\hookrightarrow Y$ of CW complexes together with a homeomorphism $\mathrm{\Phi }:(D^n,D_{+}^{n-1})\to (Q^n,Q^{n-1})$ of ball pairs and a continuous map $\phi :Q^n\to Y$ such that

• $\phi$ is a characteristic map for a new $n$-cell $e^n=\phi (Q^n\setminus \mathrm{\partial }{Q}^n)$,
• $\phi {\mathrm{\mid }}_{Q^{n-1}}$ is a characteristic map for a new $(n-1)$-cell $e^{n-1}=\phi (Q^{n-1}\setminus \mathrm{\partial }{Q}^{n-1})$,
• the remaining boundary is old: $\phi (\mathrm{\partial }{Q}^n\setminus \mathrm{int}{Q}^{n-1})\subseteq X$, and
• $Y=X\cup e^{n-1}\cup e^n$ as a CW complex, with $X$ a subcomplex.

The new $(n-1)$-cell is called the free face of the new $n$-cell. The restriction of $\phi$ to $Q^{n-1}$ maps its interior homeomorphically onto $e^{n-1}$ and maps its boundary into $X$; it need not be a homeomorphism onto the closed cell, whose attaching map may identify boundary points. The complementary boundary of $Q^n$ maps into $X$. In particular, an attachment of only one $n$-cell to a complex already containing the alleged free face is not an elementary expansion under this definition.

We say that $Y$ collapses to $X$ by an elementary collapse and write $Y\searrow X$ when $X\hookrightarrow Y$ is an elementary expansion; the elementary collapse is the inverse formal operation removing the pair $(e^{n-1},e^n)$. A finite sequence of elementary expansions and elementary collapses, each performed relative to the cells retained by the previous steps, is a formal deformation; when every cell of a subcomplex $X_0$ is retained throughout, the deformation is written relative to $X_0$, and the operations are then said to fix the retained subcomplex. The one-cell case $n=1$ attaches a new vertex and a new edge joining it to an old vertex, the new vertex being the free face.