ScalingStacks

Properties 7.3.2. Let $\gamma$ be a non-constant oriented path in $Z$.

• (1)

  We have $\gamma([0,1])\cap Z_{o}=\operatorname{supp}\nolimits([\gamma])\cap Z_{o}$ and $\gamma([0,1])\cap Z_{u}$ is contained in the union of the connected components of $Z_{u}$ that have a non-empty intersection with $\operatorname{supp}\nolimits([\gamma])$.

• (2)

  If $\gamma$ is homotopic to a constant path, then it is contained in $Z_{u}$ (as $\gamma([0,1])$ is contractible).

• (3)

  There are unique real numbers $0=t_{0}<t_{1}<\cdots<t_{r}=1$ such that

  • –

    given $0\leq i<r$, there are $\{j,k\}=\{i,i+1\}$ with the property that $\gamma([t_{j},t_{j+1}])\subseteq Z_{u}$ (if $j<r$) and $\gamma([t_{k},t_{k+1}])\subseteq\bar{Z}_{o}$ (if $k<r$) (cf Lemma 7.1.16 for $E=Z_{u}\cap\bar{Z}_{o}$).

  • –

    given $0<i<r$ and $\varepsilon>0$ such that $\gamma([t_{i},t_{i}+\varepsilon])\subset Z_{u}\cap\bar{Z}_{o}$, we have $\gamma([t_{i},t_{i+1}]){\not\subseteq}\bar{Z}_{o}$

  • –

    given $0<i<r$ and $\varepsilon>0$ such that $\gamma([t_{i}-\varepsilon,t_{i}])\subset Z_{u}\cap\bar{Z}_{o}$, we have $\gamma([t_{i-1},t_{i}]){\not\subseteq}\bar{Z}_{o}$.

  The sequence $[\gamma_{|[t_{0},t_{1}]}],\ldots,[\gamma_{|[t_{r-1},t_{r}]}]$ depends only on $[\gamma]$.

• (4)

  Consider homotopy classes of oriented paths $\zeta_{1}$, $\zeta_{2}$ and $\zeta_{3}$ with $[\gamma]=\zeta_{3}\circ\zeta_{2}\circ\zeta_{1}$. If $\operatorname{supp}\nolimits(\zeta_{2})$ is contained in $\overline{Z_{o}}$ but not in $Z_{u}$, then there are $0\leq t_{1}\leq t_{2}\leq 1$ such that $[\gamma_{|[0,t_{1}]}]=\zeta_{1}$, $[\gamma_{|[t_{1},t_{2}]}]=\zeta_{2}$ and $[\gamma_{|[t_{2},1]}]=\zeta_{3}$.

Properties 7.3.5.

• (1)

  A path is smooth if and only if its inverse is smooth.

• (2)

  A smooth path is contained in a component of $Z$.

• (3)

  Every admissible path $\gamma$ is homotopic to a minimal admissible path via a homotopy involving only admissible paths contained in the support of $\gamma$ (cf Lemma 7.1.18).

• (4)

  A minimal path in a smooth (resp. admissible) homotopy class is smooth (resp. admissible).

• (5)

  An oriented path is admissible if and only if its homotopy class is admissible (Lemma 7.1.16 provides a minimal oriented path $\gamma_{min}$ homotopic to a given oriented path $\gamma$ with the property that $\gamma$ is admissible if $\gamma_{min}$ is admissible, hence we obtain the desired equivalence by (4) above).

• (6)

  Given two oriented homotopy classes of paths $\alpha$ and $\beta$ with $\alpha\circ\beta$ admissible, then $\alpha$ and $\beta$ are admissible (cf (5) above).