89:
1 = 1 + (−1 + 1) + (−1 + 1) + ... = 1 − 1 + 1 − 1 + ... = (1 − 1) + (1 − 1) + ... = 0
295:
101:, but the analogous argument can be used in some contexts where there is some sort of "addition" defined on some objects for which infinite sums do make sense, to show that if
1112:
1187:
673:
575:
have a well-defined natural number as their dimension which is additive under direct sums, and are isomorphic if and only if they have the same dimension.
979:
1076:
676:
might be seen as antecedent of the
Eilenberg–Mazur swindle. In fact, the ideas are quite similar. If there are injections of sets from
183:
1219:
1094:
1058:
1032:
578:
This is false for some noncommutative rings, and a counterexample can be constructed using the
Eilenberg swindle as follows. Let
371:
728:, where + means disjoint union and = means there is a bijection between two sets. Expanding the former with the latter,
959:
169:
is non-trivial. For knots it is possible to take infinite sums by making the knots smaller and smaller, so if
1263:
85:
The
Eilenberg–Mazur swindle is similar to the following well known joke "proof" that 1 = 0:
1258:
410:
1214:, Mathematics Lecture Series, vol. 7, Houston, TX: Publish or Perish, Inc., pp. xiv+439,
1175:
594:(for example the direct sum of an infinite number of copies of any nonzero abelian group), and let
414:
159:
130:
47:
94:
1215:
1090:
1072:
1054:
1028:
432:
1179:
1019:, Graduate Texts in Mathematics, vol. 150, New York: Springer-Verlag, pp. xvi+785,
1154:
1145:
1121:
1020:
991:
98:
67:
39:
1242:
1229:
1200:
1168:
1133:
1042:
1005:
1225:
1196:
1164:
1129:
1038:
1001:
79:
362: + ... is Euclidean space so the Mazur swindle shows that the connected sum of
1012:
1252:
1107:
391:
126:
1207:
656:
to be the group ring of the restricted direct product of infinitely many copies of
156:
46:, is a method of proof that involves paradoxical properties of infinite sums. In
755:
consist of those elements of the left hand side that correspond to an element of
1140:
1103:
440:
51:
43:
31:
1024:
975:
996:
17:
313:
309:
640:
are any groups then the
Eilenberg swindle can be used to construct a ring
409:
In algebra the addition used in the swindle is usually the direct sum of
330:-manifolds have an addition operation given by connected sum, with 0 the
152:
134:
958:
Lam (1999), Corollary 2.7, p. 22; Eklof & Mekler (2002), Lemma 2.3,
97:
1 − 1 + 1 − 1 + ...
1159:
1125:
759:
on the right hand side. This bijection then expands to the bijection
125:
In geometric topology the addition used in the swindle is usually the
1108:"On the structure of certain semi-groups of spherical knot classes"
390:-sphere, and in some dimensions, such as 7, there are examples of
382:-sphere. (This does not show in the case of smooth manifolds that
27:
Method of proof involving paradoxical properties of infinite sums
571:, p.121) Finitely generated free modules over commutative rings
93:
This "proof" is not valid as a claim about real numbers because
308:
by a similar argument). The infinite sum of knots is usually a
290:{\displaystyle A=A+(B+A)+(B+A)+\cdots =(A+B)+(A+B)+\cdots =0\,}
943:
as well as the well-definedness of infinite disjoint union.
1017:
Commutative algebra. With a view toward algebraic geometry
397:
with inverses that are not diffeomorphic to the standard
366:
and
Euclidean space is Euclidean space, which shows that
1212:
Knots and links. Corrected reprint of the 1976 original.
186:
289:
8:
1188:Notices of the American Mathematical Society
151:in geometric topology is the proof that the
147:, chapter 4B): A typical application of the
1158:
1051:Almost free modules: set-theoretic models
995:
899:This argument depended on the bijections
286:
185:
1049:Eklof, Paul C.; Mekler, Alan H. (2002),
568:
951:
317:
144:
880:with the inverse of the bijection for
790:Substituting the right hand side for
59:
55:
7:
1143:(1961), "On embeddings of spheres",
1113:Publications Mathématiques de l'IHÉS
692:, this means that formally we have
629:
66:. In algebra it was introduced by
1180:"What is ... an infinite swindle?"
1069:Exercises in Classical Ring Theory
674:Cantor–Bernstein–Schroeder theorem
25:
980:"Big projective modules are free"
427:in algebra is the proof that if
374:of Euclidean space and therefore
598:be the ring of endomorphisms of
984:Illinois Journal of Mathematics
460:. To see this, choose a module
320:) for more geometric examples.
1245:on Mazur's swindle in topology
837:Switching every adjacent pair
582:be an abelian group such that
474:is free, which can be done as
423:A typical application of the
271:
259:
253:
241:
229:
217:
211:
199:
1:
1087:Lectures on modules and rings
876:Composing the bijection for
652:are isomorphic rings: take
644:such that the group rings
1280:
1071:, New York, NY: Springer,
610:is isomorphic to the left
62:) and is often called the
1243:Exposition by Terence Tao
1025:10.1007/978-1-4612-5350-1
386:is diffeomorphic to the
1085:Lam, Tsit-Yuen (1999),
1067:Lam, Tsit-Yuen (2003),
751:In this bijection, let
478:is projective, and put
378:is homeomorphic to the
36:Eilenberg–Mazur swindle
997:10.1215/ijm/1255637479
291:
292:
70:and is known as the
50:it was introduced by
806:gives the bijection
632:, Exercise 8.16) If
184:
109: = 0 then
76:Eilenberg telescope
1160:10.1007/BF02559532
1126:10.1007/bf02684388
287:
48:geometric topology
1176:Poénaru, Valentin
1078:978-0-387-00500-3
672:The proof of the
433:projective module
425:Eilenberg swindle
405:Eilenberg swindle
99:does not converge
72:Eilenberg swindle
16:(Redirected from
1271:
1232:
1203:
1184:
1171:
1162:
1146:Acta Mathematica
1136:
1099:
1081:
1063:
1045:
1008:
999:
963:
956:
942:
916:
719:
705:
602:. Then the left
473:
459:
439:then there is a
372:compactification
326:: The oriented
304:is trivial (and
296:
294:
293:
288:
177:is trivial then
68:Samuel Eilenberg
40:Samuel Eilenberg
21:
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1174:
1141:Mazur, Barry C.
1139:
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1013:Eisenbud, David
1011:
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918:
900:
707:
693:
670:
465:
447:
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370:is the 1-point
182:
181:
123:
95:Grandi's series
80:telescoping sum
28:
23:
22:
15:
12:
11:
5:
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1275:
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1237:External links
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1234:
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1204:
1195:(5): 619–622,
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835:
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787:
749:
748:
720:for some sets
669:
668:Other examples
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562:
561:
507:
506:
406:
403:
392:exotic spheres
346:-sphere, then
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38:, named after
26:
24:
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3:
2:
1276:
1265:
1264:Module theory
1262:
1260:
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1254:
1244:
1241:
1240:
1236:
1231:
1227:
1223:
1221:0-914098-16-0
1217:
1213:
1209:
1208:Rolfsen, Dale
1205:
1202:
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1194:
1190:
1189:
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1173:
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1156:
1153:(1–2): 1–17,
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1096:0-387-98428-3
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1060:0-444-50492-3
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1034:0-387-94268-8
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590: ⊕
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586: ≅
585:
581:
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570:
569:Eisenbud 1995
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361:
358: +
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149:Mazur swindle
146:
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127:connected sum
121:Mazur swindle
120:
118:
116:
112:
108:
105: +
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100:
96:
88:
87:
86:
83:
81:
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69:
65:
64:Mazur swindle
61:
57:
53:
49:
45:
41:
37:
33:
19:
18:Mazur swindle
1211:
1192:
1186:
1150:
1144:
1117:
1111:
1104:Mazur, Barry
1089:, Springer,
1086:
1068:
1053:, Elsevier,
1050:
1016:
987:
983:
954:
939:
935:
931:
927:
923:
919:
913:
909:
905:
901:
898:
892:
888:
884:then yields
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583:
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541:
537:
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502:
498:
494:
490:
486:
482:
475:
470:
466:
461:
456:
452:
448:
443:
436:
435:over a ring
428:
424:
420:
419:
408:
398:
394:
387:
383:
379:
375:
367:
363:
359:
355:
351:
347:
343:
339:
335:
334:-sphere. If
331:
327:
323:
322:
318:Poénaru 2007
305:
301:
299:
174:
170:
166:
162:
148:
145:Rolfsen 1990
140:
139:
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114:
110:
106:
102:
92:
84:
75:
71:
63:
35:
29:
1259:Knot theory
976:Bass, Hyman
441:free module
157:non-trivial
44:Barry Mazur
32:mathematics
1253:Categories
969:References
464:such that
401:-sphere.)
1120:: 19–27,
990:: 24–31,
684:and from
540:) ⊕ ⋯ = (
314:tame knot
312:, not a
310:wild knot
278:⋯
236:⋯
135:manifolds
1210:(1990),
1178:(2007),
1106:(1959),
1015:(1995),
978:(1963),
630:Lam 2003
614:-module
606:-module
556:) ⊕ ⋯ ≅
509:so that
421:Example:
1230:1277811
1201:2311984
1169:0125570
1134:0116347
1043:1322960
1006:0143789
845:yields
626:Example
565:Example
413:over a
411:modules
342:is the
324:Example
316:. See (
155:of two
141:Example
54: (
1228:
1218:
1199:
1167:
1132:
1093:
1075:
1057:
1041:
1031:
1004:
868:+ ⋯ +
829:+ ⋯ +
782:+ ⋯ +
34:, the
1183:(PDF)
947:Notes
930:) = (
548:) ⊕ (
532:) ⊕ (
446:with
431:is a
160:knots
131:knots
117:= 0.
78:(see
52:Mazur
1216:ISBN
1091:ISBN
1073:ISBN
1055:ISBN
1029:ISBN
960:p. 9
938:) +
917:and
724:and
706:and
648:and
636:and
505:⊕ ⋯.
415:ring
165:and
60:1961
56:1959
42:and
1155:doi
1151:105
1122:doi
1021:doi
992:doi
922:+ (
794:in
688:to
680:to
628:: (
567:: (
524:⊕ (
300:so
153:sum
133:or
129:of
82:).
74:or
30:In
1255::
1226:MR
1224:,
1197:MR
1193:54
1191:,
1185:,
1165:MR
1163:,
1149:,
1130:MR
1128:,
1116:,
1110:,
1039:MR
1037:,
1027:,
1002:MR
1000:,
986:,
982:,
934:+
926:+
912:+
908:=
904:+
891:=
864:+
860:+
856:+
852:=
841:+
825:+
821:+
817:+
813:=
802:+
798:=
778:+
774:+
770:+
766:=
743:+
739:+
735:=
715:+
711:=
701:+
697:=
664:.
622:.
552:⊕
544:⊕
536:⊕
528:⊕
520:=
516:⊕
501:⊕
497:⊕
493:⊕
489:⊕
485:=
469:⊕
455:≅
451:⊕
417:.
137:.
113:=
58:,
1157::
1124::
1118:3
1023::
994::
988:7
962:.
940:C
936:B
932:A
928:C
924:B
920:A
914:A
910:B
906:B
902:A
895:.
893:Y
889:X
882:Y
878:X
872:.
870:Z
866:B
862:A
858:B
854:A
850:Y
843:A
839:B
833:.
831:Z
827:A
823:B
819:A
815:B
811:Y
804:X
800:B
796:Y
792:X
786:.
784:Z
780:B
776:A
772:B
768:A
764:X
757:X
753:Z
747:.
745:B
741:A
737:X
733:X
726:B
722:A
717:B
713:X
709:Y
703:A
699:Y
695:X
690:X
686:Y
682:Y
678:X
662:B
658:A
654:R
650:R
646:R
642:R
638:B
634:A
620:R
616:R
612:R
608:R
604:R
600:X
596:R
592:X
588:X
584:X
580:X
573:R
560:.
558:F
554:B
550:A
546:B
542:A
538:A
534:B
530:A
526:B
522:A
518:F
514:A
503:B
499:A
495:B
491:A
487:B
483:F
476:A
471:B
467:A
462:B
457:F
453:F
449:A
444:F
437:R
429:A
399:n
395:A
388:n
384:A
380:n
376:A
368:A
364:A
360:B
356:A
352:B
348:A
344:n
340:B
336:A
332:n
328:n
306:B
302:A
284:0
281:=
275:+
272:)
269:B
266:+
263:A
260:(
257:+
254:)
251:B
248:+
245:A
242:(
239:=
233:+
230:)
227:A
224:+
221:B
218:(
215:+
212:)
209:A
206:+
203:B
200:(
197:+
194:A
191:=
188:A
175:B
171:A
167:B
163:A
143:(
115:B
111:A
107:B
103:A
20:)
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