Loop space

In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of based maps from the circle S¹ to X with the compact-open topology. Two elements of a loop space can be naturally concatenated. With this concatenation operation, a loop space is an A_∞-space. The adjective A_∞ describes the manner in which concatenating loops is homotopy coherently associative.

The quotient of the loop space ΩX by the equivalence relation of pointed homotopy is the fundamental group π₁(X).

The iterated loop spaces of X are formed by applying Ω a number of times.

An analogous construction of topological spaces without basepoint is the free loop space. The free loop space of a topological space X is the space of maps from S¹ to X with the compact-open topology. That is to say, the free loop space of a topological space X is the function space ${\mathrm {Map}}(S^{1},X)$. The free loop space of X is denoted by ${\mathcal {L}}X$.

The free loop space construction is right adjoint to the cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction can be understood as currying, where the cartesian product is adjoint to the hom functor. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. Informally, this duality, as well as currying, is referred to as Eckmann–Hilton duality.

Eckmann–Hilton duality

The loop space is dual to the suspension of the same space; this duality is sometimes called Eckmann–Hilton duality. The basic observation is that

$[\Sigma Z,X]\approxeq [Z,\Omega X]$

where $[A,B]$ is the set of homotopy classes of maps $A\rightarrow B$, and $\Sigma A$ is the suspension of A, and $\approxeq$ denotes the natural homeomorphism. This homeomprphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products.

In general, $[A,B]$ does not have a group structure for arbitrary spaces $A$ and $B$. However, it can be shown that $[\Sigma Z,X]$ and $[Z,\Omega X]$ do have natural group structures when $Z$ and $X$ are pointed, and the aforesaid isomorphism is of those groups. ^[1] Thus, setting $Z=S^{{k-1}}$ (the $k-1$ sphere) gives the relationship

$\pi _{k}(X)\approxeq \pi _{{k-1}}(\Omega X)$

This follows, since the homotopy group is defined as ${\displaystyle \pi _{k}(X)=[S^{k},X]}$, and the spheres can be obtained via suspensions of each-other: that is, ${\displaystyle S^{k}=\Sigma S^{k-1}}$. ^[2]

See also

• fundamental group
• path (topology)
• loop group
• free loop
• quasigroup
• Spectrum (topology)
• Eilenberg–MacLane space

References

1. May, J. P. (1999), A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2016-08-27 (See chapter 8, section 2)
2. Topospaces wiki - Loop space of a based topological space

• Adams, John Frank (1978), Infinite loop spaces, Annals of Mathematics Studies, 90, Princeton University Press, ISBN 978-0-691-08207-3, MR 505692
• May, J. Peter (1972), The Geometry of Iterated Loop Spaces, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0067491, ISBN 978-3-540-05904-2, MR 0420610