Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I've been given the question:

Let $ f : X \to Y $ be a continuous map, and suppose we are given (not necessarily equal) continuous maps $ g,h : Y \to X $ such that $ gf \simeq id_X $ and $ fh \simeq id_Y $. Show that $f$ is a homotopy equivalence.

What does it mean for a single function $ f: X \to Y $ to be a homotopy equivalence?


share|improve this question
It means there is a function $j:Y\rightarrow X$ so that $jf\simeq id_X$ and $fj\simeq id_Y$. –  Jason DeVito Oct 13 '11 at 16:05
add comment

1 Answer 1

Is there a specific part of the definition you don't understand? From Wikipedia:

Given two spaces $X$ and $Y$, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps $f : X → Y$ and $g : Y → X$ such that $g ∘ f$ is homotopic to the identity map id $X$ and $f ∘ g$ is homotopic to id $Y$.

The article also gives a definition of homotopy between functions

Formally, a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space X with the unit interval $[0,1]$ to $Y$ such that, if $x ∈ X$ then $H(x,0) = f(x)$ and $H(x,1) = g(x)$.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.