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Let $X$ and $Y$ be topological spaces. Let $\langle X,Y\rangle$ denote the homotopy classes of maps from $X$ and $Y$. The reduced suspension $\Sigma(-)$ has the adjoint $\Omega(-)$. In other words, we have $$ \langle \Sigma X, Y \rangle\cong \langle X,\Omega Y\rangle $$ for all $X$ and $Y$.

I am always confused with on which side I should put $\Sigma$. What is the easiest or intuitive way to think of this isomorphism? Are there a good way to memorize this this formula?

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    I think of the suspension as being made-up of a bunch of "loops" that crash through $X$. So a map out of $\Sigma X$ to $Y$ is a map from $X$ to $\Omega Y$.2012-09-24
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    I agree with Ryan. Given a map $f:\Sigma X \rightarrow Y$, one can get a loop $f_{x}:\{x\}\times I\rightarrow Y$ for each $x\in X$, where $I$ is the unit interval appearing in the definition of reduced suspension.2012-09-24

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