{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Homework 10

# Homework 10 - MATH 74 HOMEWORK 10 Due Wednesday May 6th at...

This preview shows page 1. Sign up to view the full content.

MATH 74 HOMEWORK 10 Due Wednesday May 6th at 3:10pm. (1) Prove that homotopy defines an equivalence relation on the set of based loops in a topological space. (2) Prove that multiplication in the fundamental group is well-defined. That is, given equivalence classes of loops [ γ ] and [ η ] we choose representatives γ and η so that [ γ ] * [ η ] = [ γ * η ]. If we choose different representatives, ˜ γ and ˜ η , show that [ γ ] * [ η ] = [˜ γ * ˜ η ]. HINT: There are homotopies H 1 and H 2 between γ and ˜ γ and η and ˜ η respectively. Use these to build a homotopy between γ * η and ˜ γ * ˜ η . (3) Show that following are equivalent: (a) Every map S 1 X is homotopic to the constant map.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online