Thread: Retract (algebraic topology)

Hi everyone, excuse me about my bad english but it isn't my natural language.
I have tried so much but I can´t solve the next problem:


Let be a closed orientable surface. Let a loop such that is connected. Show that is retract of .


I would like to know how to solve this.I hope sombody could help me. Thanks anyway.

Choose two distinct points . The homogeneity lemma (cf Milnor's "Topology from the Differentiable Viewpoint") guarantees that there exists a
diffeomorphism of onto itself that maps one point to the other, and that this diffeomorphism is smoothly isotopic to the identity.

cut off one of the two generator circles of a torus, the resulting space is a cylinder which is connected, while a torus does not retract to a circle since its fundamental group is Z x Z. Did I understand your question correctly?

Originally Posted by xxp9

cut off one of the two generator circles of a torus, the resulting space is a cylinder which is connected, while a torus does not retract to a circle since its fundamental group is Z x Z. Did I understand your question correctly?

A "deformation retract" and a "retract" are not the same thing. Your argument works if the question asked about a "deformation retract".

In other words, a torus does not deformation retract to a circle, but it retracts to a circle.