Misplaced Pages

In algebraic topology, a branch of mathematics, the based path space ${\displaystyle PX}$ of a pointed space ${\displaystyle (X,*)}$ is the space that consists of all maps ${\displaystyle f}$ from the interval ${\displaystyle I=}$ to X such that ${\displaystyle f(0)=*}$, called based paths. In other words, it is the mapping space from ${\displaystyle (I,0)}$ to ${\displaystyle (X,*)}$.

A space ${\displaystyle X^{I}}$ of all maps from ${\displaystyle I}$ to X, with no distinguished point for the start of the paths, is called the free path space of X. The maps from ${\displaystyle I}$ to X are called free paths. The path space ${\displaystyle PX}$ is then the pullback of ${\displaystyle X^{I}\to X,\,\chi \mapsto \chi (0)}$ along ${\displaystyle *\hookrightarrow X}$.

The natural map ${\displaystyle PX\to X,\,\chi \to \chi (1)}$ is a fibration called the path space fibration.

See also

• mapping space

References

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 (PDF). Graduate Studies in Mathematics. Vol. 35. Providence, RI: American Mathematical Society. pp. xvi+367. doi:10.1090/gsm/035. ISBN 0-8218-2160-1. MR 1841974.

Further reading

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

This topology-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Topology stubs
• Algebraic topology