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(Redirected from Bouquet of circles)
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.
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'.
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.
★ 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.
★ Free group
★ Topological graph
★ Hawaiian earring
★ Algebraic topology, Hatcher, Allen, , , Cambridge University Press, 2002,
★ Topology, Munkres, James R., , , Prentice Hall, Inc, 2000,
★ Classical topology and combinatorial group theory, Stillwell, John, , , Springer-Verlag, 1993,
A rose with four petals.
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.
| Contents |
| Definition |
| Relation to free groups |
| Other properties |
| See also |
| References |
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 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 is the Cayley graph of the free group.
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.
★ 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
★ Algebraic topology, Hatcher, Allen, , , Cambridge University Press, 2002,
★ Topology, Munkres, James R., , , Prentice Hall, Inc, 2000,
★ Classical topology and combinatorial group theory, Stillwell, John, , , Springer-Verlag, 1993,
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