---- > [!definition] Definition. ([[suspension of a topological space]]) > The **suspension** $\Sigma X$ of a [[topological space]] $X$ is the [[quotient space|quotient]] of $X \times [0,1]$ be the [[equivalence relation]] which collapses each end of the cylinder to a point: $X \times \{ 0 \} \sim p_{0}$ and $X \times \{ 1 \} \sim p_{1}$. > Any map $f:X \to Y$ **suspends** to a map $\Sigma f:\Sigma X \to \Sigma Y$ via the [[universal property of quotient topology]] applied to $\pi_{Y} \circ f$: > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAA0ACAHW7wFt4nAJIgAvqXSZc+QigCM5KrUYs2ATR58sguCPGSQGbHgJEy85fWatEIXgGUsAc351O7A1JOyiiy9TWanaOLm6c6uLKMFDO8ESgAGYAThD8SGQgOBBIAEyBqrb23GhYAPqe1Ax0AEYwDAAK0qZyIMkuABY4XiApaUiKWTmIAMwFNmy8pWWREkmp6Yj5Q0hjIAxYYEVQEDg4MSDjwcUwAB5YcDh6AIScWk6u7omH67X1TT5mdu3OXT19i0y2QGRyKiS0AiEojEFDEQA > \begin{tikzcd} > X \times I \arrow[d, "\pi_X"'] \arrow[r, "f \times \text{id}_I"] & Y \times I \arrow[d, "\pi_Y"] \\ > \Sigma X \arrow[r, "\exists ! \Sigma f"', dotted] & \Sigma Y > \end{tikzcd} > \end{document} > ``` > That is, $\Sigma f([x, t])=\pi_{Y} \circ f\times \id_{I}(x, t)$. Explicitly: > $\Sigma f([x,t]) = \begin{cases} p_0 & \text{if } t = 0 \\ [f(x),t] & \text{if } 0 < t < 1 \\ p_1 & \text{if } t = 1 \end{cases}$ > > - [ ] picture > [!basicexample] > - $\Sigma \mathbb{S}^{n} \cong \mathbb{S}^{n+1}$. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```