ScalingStacks

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).