Rose (topology)

A rose with four petals.

In mathematics, a rose (also known as a bouquet of n 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 are closely related to free groups.

Definition

The fundamental group of the figure eight is the free group generated by a and b

A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of 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 universal cover of the figure eight can be visualized by the Cayley graph of the free group on two generators a and b

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

A figure eight in the torus.

See also

References

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