---
tag: Q7RFNJH76Q
kind: concept
name: Suspension
msc: [55P40, 54B15]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Suspension_(topology)
same_as:
  wikidata: Q1307987
  wikipedia: Suspension (topology)
---

## Summary

The suspension $SX$ of a topological space $X$ is the cylinder $X \times [0, 1]$ with each end collapsed to its own point.
It looks like two cones on $X$ glued along their bases.
The suspension of the $n$-sphere is the $(n+1)$-sphere, and suspension shifts reduced homology up by one degree.
It is a basic construction in algebraic topology and stable homotopy theory.

## Explanations

### L1 human

Take the space $X$ and stretch it into a tube $X \times [0, 1]$.
Then pinch the whole bottom end to one point and the whole top end to another point.
The space now hangs between a south pole and a north pole.
For example, the suspension of two points is a circle: each point becomes a path from the south pole to the north pole.
The suspension of a circle is a sphere, like a globe with its meridians.