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loop (Definition)

A loop based at $ x_0$ in a topological space $ X$ is simply a continuous map $ f : [0,1]\to X$ with $ f(0) = f(1) = x_0$.

The collection of all such loops, modulo homotopy equivalence, forms a group known as the fundamental group.

More generally, the space of loops in $ X$ based at $ x_0$ with the compact-open topology, represented by $ \Omega_{x_0}$, is known as the loop space of $ X$. And one has the homotopy groups $ \pi_n(X,x_0) = \pi_{n-1}(\Omega_{x_0},\iota)$, where $ \pi_n$ represents the higher homotopy groups, and $ \iota$ is the basepoint in $ \Omega_{x_0}$ consisting of the constant loop at $ x_0$.



"loop" is owned by nerdy2.
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Cross-references: basepoint, represents, homotopy groups, loop space, compact-open topology, fundamental group, group, homotopy equivalence, collection, continuous map, topological space
There are 12 references to this entry.

This is version 2 of loop, born on 2002-02-03, modified 2002-07-24.
Object id is 1706, canonical name is Loop.
Accessed 3593 times total.

Classification:
AMS MSC: 54-00 (General topology :: General reference works )
Pending Errata and Addenda
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