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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
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with one point removed deformation retracts onto a figure eight, namely the union of two generating circles. More generally, a surface of
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431:: there is a continuous bijection from this rose onto the Hawaiian earring, but the two are not
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435:. A rose with infinitely many petals is not compact, whereas the Hawaiian earring is compact.
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Topological space obtained by gluing together a collection of circles along a single point
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provides a simple proof that every subgroup of a free group is free (the
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points on a single circle. The rose with two petals is known as the
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petals. One petal of the rose surrounds each of the removed points.
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of the free group. The observation that any cover of a rose is a
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of the figure eight can be visualized by the Cayley graph of the
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with one point removed deformation retracts onto a rose with 2
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along a single point. The circles of the rose are called
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is an infinite tree, which can be identified with the
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a set consisting of one point from each circle. As a
280:of the free group. (This is a special case of the
529:Classical topology and combinatorial group theory
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57:but its sources remain unclear because it lacks
513:, Englewood Cliffs, N.J: Prentice Hall, Inc,
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221:petals can also be obtained by identifying
367:to a rose. Specifically, the rose is the
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310:Because the universal cover of a rose is
88:Learn how and when to remove this message
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371:of the graph obtained by collapsing a
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187:. That is, the rose is the
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531:, Berlin: Springer-Verlag,
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164:of the figure eight is the
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314:, the rose is actually an
295:of the rose correspond to
140:. Roses are important in
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527:Stillwell, John (1993),
305:Nielsen–Schreier theorem
104:A rose with four petals.
43:This article includes a
316:Eilenberg–MacLane space
286:presentation of a group
233:Relation to free groups
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350:A figure eight in the
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272:for each petal. The
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396:deformation retracts
282:presentation complex
422:fundamental polygon
365:homotopy equivalent
560:Algebraic topology
555:Topological spaces
489:Algebraic topology
456:List of topologies
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334:) are trivial for
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249:on two generators
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142:algebraic topology
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45:list of references
18:Figure-eight space
507:Munkres, James R.
471:Topological graph
398:onto a rose with
291:The intermediate
262:fundamental group
212:topological graph
162:fundamental group
126:topological space
116:(also known as a
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16:(Redirected from
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461:Petal projection
429:Hawaiian earring
342:Other properties
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64:Please help
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146:free groups
118:bouquet of
110:mathematics
70:introducing
549:Categories
477:References
451:Free group
324:cohomology
247:free group
166:free group
152:Definition
297:subgroups
270:generator
181:wedge sum
78:June 2017
511:Topology
509:(2000),
486:(2002),
440:See also
198:, where
326:groups
185:circles
134:circles
124:) is a
122:circles
66:improve
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293:covers
138:petals
130:gluing
411:genus
407:torus
390:with
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51:, or
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515:ISBN
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359:Any
266:free
260:The
253:and
241:The
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