# Rose (topology)

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

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 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 *H ^{n}*(

*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 countably infinitely many petals is similar to the Hawaiian earring: there is a continuous bijection from this rose onto the Hawaiian earring, but the two are not homeomorphic.

## References

- Hatcher, Allen (2002),
*Algebraic topology*, Cambridge, UK: Cambridge University Press, ISBN 0-521-79540-0 - Munkres, James R. (2000),
*Topology*, Englewood Cliffs, N.J: Prentice Hall, Inc, ISBN 0-13-181629-2 - Stillwell, John (1993),
*Classical topology and combinatorial group theory*, Berlin: Springer-Verlag, ISBN 0-387-97970-0