Rose (topology)

In mathematics, a rose (also known as a bouquet of circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they closely related to free groups.

Definition

A rose is a wedge sum of circles. That is, the rose is the quotient space "C"/"S", where "C" is a disjoint union circles and "S" a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for each circle. This makes it a simple example of a topological graph.

A rose with "n" petals can also be obtained by identifying "n" points on a single circle. The rose with two petals is known as the figure eight.

Relation to free groups

The fundamental group of a rose is free, with one generator for each petal. The universal cover is an infinite tree, which can be identified with the Cayley graph of the free group. (This is a special case of the presentation complex associated to any presentation of a group.)

The intermediate covers of the rose correspond to subgroups of the free group. The observation that any cover of a rose is a graph provides a simple proof that every subgroup of a free group is free (the Nielsen-Schreier Theorem).

Because the universal cover of a rose is contractible, the rose is actually an Eilenberg-MacLane space for the associated free group "F". This implies that the cohomology groups "Hⁿ"("F") are trivial for "n" ≥ 2.

Other properties

* Any connected graph is homotopy equivalent to a rose. Specifically, the rose is the quotient space of the graph obtained by collapsing a spanning tree.
* A disc with "n" points removed (or a sphere with "n" + 1 points removed) deformation retracts onto a rose with "n" petals. One petal of the rose surrounds each of the removed points.
* A torus with one point removed deformation retracts onto a figure eight, namely the union of two generating circles. More generally, a surface of genus "g" with one point removed deformation retracts onto a rose with 2"g" petals, namely the boundary of a fundamental polygon.
* A rose can have infinitely many petals, leading to a fundamental group which is free on infinitely many generators. The rose with infinitely many petals is similar to (but not homeomorphic with) the Hawaiian earring.

See also

* Free group
* Topological graph
* Hawaiian earring

References

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