ScalingStacks

$$\begin{array}[]{ccl}X_{1}&=&\{x+\tau_{1}(x)+y|x\in\overline{C}(D_{1}(\ast 100))[-1]\{-1\},y\in\overline{C}(D_{1}(\ast 1))[-1]\{-1\}\}\\ X_{2}&=&\{x+d_{1}y|x,y\in\overline{C}(D_{1}(\ast 000))\}\\ X_{3}&=&\{\delta_{1}(x)+d_{1}\delta_{1}(y)|x,y\in\overline{C}(D_{1}(\ast 110))[-2]\{-2\}\}\end{array}$$

(125)

where $d_{1}$ denotes the differential of $\overline{C}(D_{1}).$ Warning: These $X_{1},X_{2},X_{3}$ have no relation to the complexes $X_{1},X_{2},X_{3}$ considered in Section 5.3.