In mathematics, a rose (also known as a bouquet of ''n'' circles) is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

obtained by gluing together a collection of

circles A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

along a single point. The circles of the rose are called petals. Roses are important in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

, where they are closely related to

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1 ...

Definition

A rose is a

wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...

. 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...

of a rose is

free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...

, with one generator for each petal. The

universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...

is an infinite tree, which can be identified with the

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayl ...

of the free group. (This is a special case of the presentation complex associated to any

presentation of a group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...

.) The intermediate covers of the rose correspond to

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

s of the free group. The observation that any cover of a rose is a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...

provides a simple proof that every subgroup of a free group is free (the

Nielsen–Schreier theorem In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier. Statement of the theorem A free group may be defined from a gro ...

) Because the universal cover of a rose is

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...

, the rose is actually an

Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...

for the associated free group ''F''. This implies that the

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

groups ''Hⁿ''(''F'') are trivial for ''n'' ≥ 2.

Other properties

* Any

connected graph In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgr ...

homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...

to a rose. Specifically, the rose is the quotient space of the graph obtained by collapsing a

spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not ...

. * A disc with ''n'' points removed (or a

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

with ''n'' + 1 points removed)

deformation retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...

s onto a rose with ''n'' petals. One petal of the rose surrounds each of the removed points. * A

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

with one point removed deformation retracts onto a figure eight, namely the union of two generating circles. More generally, a surface of

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nomencla ...

''g'' with one point removed deformation retracts onto a rose with 2''g'' petals, namely the boundary of a

fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equ ...

. * 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 In mathematics, the Hawaiian earring \mathbb is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac,0\right) and radius \tfrac for n = 1, 2, 3, \ldots endowed with the subspace topology: ...

: there is a continuous bijection from this rose onto the Hawaiian earring, but the two are not

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...

. A rose with infinitely many petals is not compact, whereas the Hawaiian earring is compact.

References

* * * {{citation , last=Stillwell , first= John , title=Classical topology and combinatorial group theory , publisher=Springer-Verlag , location=Berlin , year=1993 , isbn=0-387-97970-0 Topological spaces Algebraic topology

Definition

Relation to free groups

Other properties

See also

References