Consider [itex]h:S^1 \times I \to S^1[/itex] defined by [itex]h(s,t) = s^t[/itex]. This is a homotopy from the constant map 1 to the identity function on [itex]S^1[/itex]. On the other hand, it's easy to show that degree is homotopy invariant. However, we have [itex]\mathrm{deg}(1) = 0[/itex] while [itex]\mathrm{deg}(id) = 1[/itex]. Clearly, I've got something mixed up with how I'm thinking about homotopies. Can anybody shed some light on my error?(adsbygoogle = window.adsbygoogle || []).push({});

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Confusion About Homotopies

**Physics Forums | Science Articles, Homework Help, Discussion**