Semi-locally simply connected

Short description: Property in algebraic topology

In mathematics, specifically algebraic topology, the semi-locally simply connected (or semilocally simply connected) property is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space X satisfies the property if for each point x in X any sufficiently small loop going through x can be contracted within X to a point. This condition is necessary for most of the theory of covering spaces, including the existence of a universal cover and the Galois correspondence between covering spaces and subgroups of the fundamental group.

Most “nice” spaces such as manifolds and CW complexes are semi-locally simply connected, and topological spaces that do not satisfy this condition are considered somewhat pathological. The standard example of a non-semi-locally simply connected space is the Hawaiian earring.

Definition

A space X is called semi-locally simply connected if every point x in X has a neighborhood U with the property that every loop in U based at x can be contracted within X to the constant loop at x (i.e., every loop in U starting and ending at x is nullhomotopic in X via a basepoint-preserving homotopy).

Note that if U satisfies this condition, so does any smaller neighborhood of x, so that x has arbitrarily small neighborhoods satisfying the condition. The neighborhood U need not be simply connected: though every loop in U based at x must be contractible within X, the contraction is not required to take place inside of U. For this reason, a space can be semi-locally simply connected without being locally simply connected. Also, it is not required that every loop in U is nullhomotopic in X; it is only the loops in U based at x that must be nullhomotopic in X. In general, a semi-locally simply connected space may have points x with arbitrarily small neighborhoods containing loops (not going through x) that cannot be contracted to a point, even with homotopies in X.

An equivalent formulation of the definition is that every point in $x\in X$ has an open neighborhood $U$ for which the homomorphism ${\pi }_1(U,x)\to {\pi }_1(X,x)$ induced by the inclusion map of $U$ into $X$ is trivial. Here, ${\pi }_1(U,x)$ is the fundamental group of $U$ relative to the basepoint $x;$ and similarly for ${\pi }_1(X,x).$

Most of the main theorems about covering spaces, including the existence of a universal cover and the Galois correspondence, require a space to be path-connected, locally path-connected, and semi-locally simply connected, a condition known as unloopable (délaçable in French). In particular, this condition is necessary for a locally path-connected space to have a simply connected covering space.

Examples

The Hawaiian earring is not semi-locally simply connected.

A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number. Give this space the subspace topology. Then all neighborhoods of the origin contain circles that are not nullhomotopic.

The Hawaiian earring can also be used to construct a semi-locally simply connected space that is not locally simply connected. In particular, the cone on the Hawaiian earring is contractible and therefore semi-locally simply connected, but it is clearly not locally simply connected.

References

• Bourbaki, Nicolas (2016). Topologie algébrique: Chapitres 1 à 4. Springer. Ch. IV pp. 339 -480. ISBN 978-3662493601. 
• J.S. Calcut, J.D. McCarthy Discreteness and homogeneity of the topological fundamental group Topology Proceedings, Vol. 34,(2009), pp. 339–349
• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. https://pi.math.cornell.edu/~hatcher/AT/ATpage.html. 
• Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9.