Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the path space PX of a based space (X,*) is the space that consists of all maps f from the interval I=[0,1] to X such that f(0)=*, called paths.[1] In other words, it is the mapping space from (I,0) to (X,*). The space XI of all maps from I to X (free paths or just paths) is called the free path space of X.[2] The path space PX can then be viewed as the pullback of XI->X,χ↦χ(0) along *↪X.[1]

The natural map PX->X,χ->χ(1) is a fibration called the path space fibration.[3]

## References

1. ↑ 1.0 1.1 Martin Frankland, Math 527 - Homotopy Theory - Fiber sequences
2. ↑ Davis & Kirk 2001, Definition 6.14.
3. ↑ Davis & Kirk 2001, Theorem 6.15. 2.

* Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology. Graduate Studies in Mathematics. 35. Providence, RI: American Mathematical Society. pp. xvi+367. doi:10.1090/gsm/035. ISBN 0-8218-2160-1. http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf.

## Further reading

* https://ncatlab.org/nlab/show/path+space

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Path space (algebraic topology). Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Path_space_(algebraic_topology)&oldid=2523424"