25:
227:
146:
90:
336:
416:
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
54:
398:
with one point removed deformation retracts onto a figure eight, namely the union of two generating circles. More generally, a surface of
525:
507:
486:
76:
47:
293:
548:
543:
361:
349:
304:
357:
258:
177:
118:
37:
41:
33:
274:
199:, a rose has a single vertex, and one edge for each circle. This makes it a simple example of a
420:: there is a continuous bijection from this rose onto the Hawaiian earring, but the two are not
58:
424:. A rose with infinitely many petals is not compact, whereas the Hawaiian earring is compact.
270:
476:
16:
Topological space obtained by gluing together a collection of circles along a single point
8:
410:
399:
384:
353:
444:
368:
300:
130:
521:
503:
482:
459:
289:
250:
200:
173:
150:
122:
114:
449:
417:
312:
262:
231:
281:
537:
495:
472:
434:
454:
421:
266:
196:
292:
provides a simple proof that every subgroup of a free group is free (the
98:
439:
254:
235:
154:
145:
134:
226:
214:
points on a single circle. The rose with two petals is known as the
169:
391:
petals. One petal of the rose surrounds each of the removed points.
285:
89:
288:
of the free group. The observation that any cover of a rose is a
234:
of the figure eight can be visualized by the Cayley graph of the
376:
335:
405:
with one point removed deformation retracts onto a rose with 2
395:
340:
125:
along a single point. The circles of the rose are called
265:
is an infinite tree, which can be identified with the
195:
a set consisting of one point from each circle. As a
269:of the free group. (This is a special case of the
518:Classical topology and combinatorial group theory
535:
46:but its sources remain unclear because it lacks
502:, Englewood Cliffs, N.J: Prentice Hall, Inc,
481:, Cambridge, UK: Cambridge University Press,
210:petals can also be obtained by identifying
356:to a rose. Specifically, the rose is the
221:
515:
299:Because the universal cover of a rose is
77:Learn how and when to remove this message
334:
225:
144:
88:
494:
471:
536:
360:of the graph obtained by collapsing a
133:, where they are closely related to
18:
330:
191:is a disjoint union of circles and
13:
14:
560:
409:petals, namely the boundary of a
23:
383: + 1 points removed)
307:for the associated free group
1:
465:
140:
176:. That is, the rose is the
7:
520:, Berlin: Springer-Verlag,
428:
153:of the figure eight is the
10:
565:
303:, the rose is actually an
284:of the rose correspond to
129:. Roses are important in
311:. This implies that the
121:together a collection of
516:Stillwell, John (1993),
294:Nielsen–Schreier theorem
93:A rose with four petals.
32:This article includes a
305:Eilenberg–MacLane space
275:presentation of a group
222:Relation to free groups
61:more precise citations.
344:
339:A figure eight in the
246:
165:
94:
375:points removed (or a
338:
261:for each petal. The
229:
148:
92:
385:deformation retracts
271:presentation complex
411:fundamental polygon
354:homotopy equivalent
549:Algebraic topology
544:Topological spaces
478:Algebraic topology
445:List of topologies
345:
323:) are trivial for
273:associated to any
247:
238:on two generators
166:
131:algebraic topology
95:
34:list of references
496:Munkres, James R.
460:Topological graph
387:onto a rose with
280:The intermediate
251:fundamental group
201:topological graph
151:fundamental group
115:topological space
105:(also known as a
87:
86:
79:
556:
530:
512:
491:
450:Petal projection
418:Hawaiian earring
331:Other properties
327: ≥ 2.
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
564:
563:
559:
558:
557:
555:
554:
553:
534:
533:
528:
510:
489:
468:
431:
350:connected graph
333:
263:universal cover
232:universal cover
224:
143:
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
562:
552:
551:
546:
532:
531:
526:
513:
508:
492:
487:
473:Hatcher, Allen
467:
464:
463:
462:
457:
452:
447:
442:
437:
430:
427:
426:
425:
414:
392:
365:
358:quotient space
332:
329:
223:
220:
178:quotient space
142:
139:
85:
84:
42:external links
31:
29:
22:
15:
9:
6:
4:
3:
2:
561:
550:
547:
545:
542:
541:
539:
529:
527:0-387-97970-0
523:
519:
514:
511:
509:0-13-181629-2
505:
501:
497:
493:
490:
488:0-521-79540-0
484:
480:
479:
474:
470:
469:
461:
458:
456:
453:
451:
448:
446:
443:
441:
438:
436:
435:Bouquet graph
433:
432:
423:
419:
415:
412:
408:
404:
401:
397:
393:
390:
386:
382:
378:
374:
370:
366:
363:
362:spanning tree
359:
355:
351:
347:
346:
342:
337:
328:
326:
322:
318:
314:
310:
306:
302:
297:
295:
291:
287:
283:
278:
276:
272:
268:
264:
260:
256:
253:of a rose is
252:
245:
241:
237:
233:
228:
219:
217:
213:
209:
204:
202:
198:
194:
190:
186:
182:
179:
175:
171:
164:
160:
157:generated by
156:
152:
147:
138:
136:
132:
128:
124:
120:
116:
112:
110:
104:
100:
91:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
517:
499:
477:
455:Quadrifolium
422:homeomorphic
406:
402:
388:
380:
372:
324:
320:
316:
308:
301:contractible
298:
279:
267:Cayley graph
248:
243:
239:
216:figure eight
215:
211:
207:
206:A rose with
205:
197:cell complex
192:
188:
184:
180:
168:A rose is a
167:
162:
158:
126:
117:obtained by
108:
106:
102:
96:
73:
64:
53:Please help
45:
257:, with one
135:free groups
107:bouquet of
99:mathematics
59:introducing
538:Categories
466:References
440:Free group
313:cohomology
236:free group
155:free group
141:Definition
286:subgroups
259:generator
170:wedge sum
67:June 2017
500:Topology
498:(2000),
475:(2002),
429:See also
187:, where
315:groups
174:circles
123:circles
113:) is a
111:circles
55:improve
524:
506:
485:
377:sphere
282:covers
127:petals
119:gluing
400:genus
396:torus
379:with
371:with
341:torus
290:graph
40:, or
522:ISBN
504:ISBN
483:ISBN
369:disc
348:Any
255:free
249:The
242:and
230:The
161:and
149:The
103:rose
101:, a
352:is
277:.)
172:of
97:In
540::
394:A
367:A
296:)
218:.
203:.
137:.
44:,
36:,
413:.
407:g
403:g
389:n
381:n
373:n
364:.
343:.
325:n
321:F
319:(
317:H
309:F
244:b
240:a
212:n
208:n
193:S
189:C
185:S
183:/
181:C
163:b
159:a
109:n
80:)
74:(
69:)
65:(
51:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
