Weakly contractible

In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial. Equivalently, a space is weakly contractible if it is weakly homotopy equivalent to a point.

Properties

Every contractible space is weakly contractible; conversely, it follows from Whitehead's theorem that every weakly contractible CW-complex is contractible. For general topological spaces only the former implication holds.

Every weakly contractible space is in particular path-connected, simply connected and aspherical.

Examples

Define ${\displaystyle S^{\infty }}$ to be the inductive limit of the spheres ${\displaystyle S^{n},n\geq 1}$. Then this space is weakly contractible. Since ${\displaystyle S^{\infty }}$ is moreover a CW-complex, it is also contractible. See Contractibility of unit sphere in Hilbert space for more.

The long line is an example of a space which is weakly contractible, but not contractible. This does not contradict Whitehead's theorem since the long line does not have the homotopy type of a CW-complex. Another prominent example for this phenomenon is the Warsaw circle.

References

• "Homotopy type", Encyclopedia of Mathematics , EMS Press, 2001 [1994]