Supplement II (Ex)

Let $K \subseteq L$$<wikitex>; Let $K \subseteq L$ be a simplicial subcomplex. The supplement is a subcomplex $\overline{K} \subseteq L'$. Construct an embedding $|L| \subset |K'| \ast |\overline K|$ into the join of the two realizations. A point in $|K'| \ast |\overline K|$ can be described as $t \cdot x + (1-t) \cdot y$ for $x \in |K'|$, $y \in |\overline K|$, and $t \in [0,1]$. The space $|L|$ can be decomposed as the union of two subspaces $$ N = N(K') = \{ t \cdot x + (1-t) \cdot y \; | \; t \geq 1/2 \} \cap L $$ and $$ \overline N = N(\overline K) = \{ t \cdot x + (1-t) \cdot y \; | \; t \leq 1/2 \} \cap L. $$ Show that there are deformation retractions $r \colon N \rightarrow |K|$ and $\overline r \colon \overline N \rightarrow |\overline K|$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]K \subseteq L$ be a simplicial subcomplex. The supplement is a subcomplex $\overline{K} \subseteq L'$$\overline{K} \subseteq L'$. Construct an embedding $|L| \subset |K'| \ast |\overline K|$$|L| \subset |K'| \ast |\overline K|$ into the join of the two realizations. A point in $|K'| \ast |\overline K|$$|K'| \ast |\overline K|$ can be described as $t \cdot x + (1-t) \cdot y$$t \cdot x + (1-t) \cdot y$ for $x \in |K'|$$x \in |K'|$, $y \in |\overline K|$$y \in |\overline K|$, and $t \in [0,1]$$t \in [0,1]$. The space $|L|$$|L|$ can be decomposed as the union of two subspaces

$\displaystyle N = N(K') = \{ t \cdot x + (1-t) \cdot y \; | \; t \geq 1/2 \} \cap L$ $$\displaystyle N = N(K') = \{ t \cdot x + (1-t) \cdot y \; | \; t \geq 1/2 \} \cap L$$

and

$\displaystyle \overline N = N(\overline K) = \{ t \cdot x + (1-t) \cdot y \; | \; t \leq 1/2 \} \cap L.$ $$\displaystyle \overline N = N(\overline K) = \{ t \cdot x + (1-t) \cdot y \; | \; t \leq 1/2 \} \cap L.$$

Show that there are deformation retractions $r \colon N \rightarrow |K|$$r \colon N \rightarrow |K|$ and $\overline r \colon \overline N \rightarrow |\overline K|$$\overline r \colon \overline N \rightarrow |\overline K|$.