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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex The circles of the rose are called petals. Roses are important in algebraic topology, where they closely related to free groups. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be
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Definition
A rose is a wedge sum of circles. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be In Topology, the wedge sum (sometimes wedge product, though not to be confused with the Exterior product, which also shares this terminology is a "one-point In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex 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. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying As a cell complex, a rose has a single vertex, and one edge for each circle. In Topology, a CW complex is a type of Topological space introduced by J This makes it a simple example of a topological graph. In Mathematics topological graph theory is a branch of Graph theory.
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. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the The universal cover is an infinite tree, which can be identified with the Cayley graph of the free group. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group. (This is a special case of the presentation complex associated to any presentation of a group. In Geometric group theory, a presentation complex is a 2-dimensional Cell complex associated to any presentation of a group G. In Mathematics, one method of defining a group is by a presentation. )
The intermediate covers of the rose correspond to subgroups of the free group. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of 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). In Mathematics topological graph theory is a branch of Graph theory.
Because the universal cover of a rose is contractible, the rose is actually an Eilenberg-MacLane space for the associated free group F. In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i In Mathematics, an Eilenberg-MacLane space is a special kind of Topological space that can be regarded as a building block for Homotopy theory. This implies that the cohomology groups Hn(F) are trivial for n ≥ 2. In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper
Other properties
- Any connected graph is homotopy equivalent to a rose. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Mathematics topological graph theory is a branch of Graph theory. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical Specifically, the rose is the quotient space of the graph obtained by collapsing a spanning tree. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In the mathematical field of Graph theory, a spanning tree T of a connected, Undirected graph G is a tree composed
- A disc with n points removed (or a sphere with n + 1 points removed) deformation retracts onto a rose with n petals. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar More generally, a surface of genus g with one point removed deformation retracts onto a rose with 2g petals, namely the boundary of a fundamental polygon. In Mathematics, genus has a few different but closely related meanings Topology Orientable surface In Mathematics, each closed Surface in the sense of Geometric topology can be constructed from an even-sided oriented Polygon, called a fundamental
- 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. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, the Hawaiian earring H is the Topological space defined by the union of circles in the Euclidean plane R^2
See also
References
- Hatcher, Allen (2002), Algebraic topology, Cambridge, UK: Cambridge University Press, ISBN 0-521-79540-0
- Munkres, James R. (2000), Topology, Englewood Cliffs, N. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be In Mathematics topological graph theory is a branch of Graph theory. In Mathematics, the Hawaiian earring H is the Topological space defined by the union of circles in the Euclidean plane R^2 Allen Edward Hatcher is an American Topologist and also a noted author James Raymond Munkres is a Professor Emeritus of Mathematics at MIT and the author of several texts in the area of Topology, including Topology 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
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