0
$\begingroup$

This is very bizzare question, but still concerns me.

Is every path homotopic to its inverse; where by inverse I mean the same path with the opposite orientation?

If so, what can I use for homotopy?

  • 0
    If you allow your homotopy to move end points of the path, sure. Just compose your path with a homotopy that maps $[0,1]$ to "$[1, 0]$". If not, then how could they be homotopic? – 2017-01-14
  • 0
    Does this mean that, if my homotopy is relative to the end points, they are not homotopic, except for the case of loops? – 2017-01-14
  • 0
    In that case, only _nullhomotopic_ loops are homotopic relative to their inverse. – 2017-01-14
  • 2
    Actually the generator of $\pi_1(\mathbb{R}p(2))$ is not trivial and is homotopic to its inverse. – 2017-01-15

1 Answers 1

0

I will consider loops instead of paths just to show the possible scenarios. There are two things to consider.

1) Homotopy fixing base point: In this you question reduces to the question "When an element is equal to its inverse?" We clearly for $\mathbb{Z}_2$ this is true. So for any space with fundamental group $\mathbb{Z}_2$ (e.g. $\mathbb{R}\mathbb{P}^2$) this is true. For groups like $\mathbb{Z}$ this is not true.

2) Homotopy without base point. This is called free homotopy. Here the free homotopy classes corresponds to conjugacy classes. Hence your question can be interpreted as "When and element is conjugate to its inverse?". Again this depends on the group. For fundamental group of surface of genus $g>2$, this never happens.