Free loop

In the mathematical field of topology, a free loop is a variant of the mathematical notion of a loop. Whereas a loop has a distinguished point on it, called a basepoint, a free loop lacks such a distinguished point. Formally, let be a topological space. Then a free loop in is an equivalence class of continuous functions from the circle to . Two loops are equivalent if they differ by a reparameterization of the circle. That is, if there exists a homeomorphism such that .

Thus, a free loop, as opposed to a based loop used in the definition of the fundamental group, is a map from the circle to the space without the basepoint-preserving restriction. Free homotopy classes of free loops correspond to conjugacy classes in the fundamental group.

In the last years, interest in the space of all free loops has grown with the advent of string topology, i.e. the study of new algebraic structures on the homology of the free loop space.

See also

Further reading

Free Loop (One Night Stand)

"Free Loop (One Night Stand)" (titled as "Free Loop" on Daniel Powter) is a pop song written by Canadian singer Daniel Powter. It was his second single and the follow-up to his successful song, "Bad Day". It did not achieve the success of its predecessor. In the UK, WEA failed to realize the single was deemed ineligible to chart since "Bad Day" was included as B-side whilst still being in the UK Top 40 at the time. In the U.S., it did not chart on the Billboard Hot 100, but it did crack the Top 30 of Billboard's Adult Contemporary chart.

Music video

The video, directed by Marc Webb, who also directed the music videos for "Bad Day" and "Lie to Me," shows a young Daniel Powter starting at a piano in a music store and puts his fingers on the window and magically plays the piano. The scene then cuts to an adult version of Powter, who is now able to buy the piano. He then sees a note of eviction on his apartment door and has to sell the piano back to pay for the rent. After selling the piano back, he is shown at a bar where a talent scout talks to him about being part of a once-in-a-lifetime offer. Getting ready for his "big performance," he sees the piano demolished after being used in a strip club with glasses thrown at and prostitutes landing on the keys. He then takes it home and repairs it to its former glory. At the final scene he performs it at the "Billmore Hotel," and gets an applause from crowd at the end of the song. All the scenes were filmed in California.