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. 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. 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 2g petals, namely the boundary of a fundamental polygon.

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Rose (topology)

Hawaiian earring

In mathematics, the Hawaiian earring is the topological space defined by the union of circles in the Euclidean plane with center and radius for endowed with the subspace topology: The space is homeomorphic to the one-point compactification of the union of a countable family of disjoint open intervals. The Hawaiian earring is a one-dimensional, compact, locally path-connected metrizable space. Although is locally homeomorphic to at all non-origin points, is not semi-locally simply connected at .

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 and ) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification where is the equivalence closure of the relation More generally, suppose is a indexed family of pointed spaces with basepoints The wedge sum of the family is given by: where is the equivalence closure of the relation In other words, the wedge sum is the joining of several spaces at a single point.

Cellular structure of the torus with two holes MATH-225: Topology

Explores the cellular structure of the torus with two holes.