780:
express that dependence, with the RHS sum being the case where there is no dependence. This definition of codimension in terms of the number of functions needed to cut out a subspace extends to situations in which both the ambient space and subspace are infinite dimensional.
591:
888:. In fact, the theory of high-dimensional manifolds, which starts in dimension 5 and above, can alternatively be said to start in codimension 3, because higher codimensions avoid the phenomenon of knots. Since
401:
308:
217:
899:
in codimension 2 is knot theory, and difficult, while the study of embeddings in codimension 3 or more is amenable to the tools of high-dimensional geometric topology, and hence considerably easier.
469:
892:
requires working up to the middle dimension, once one is in dimension 5, the middle dimension has codimension greater than 2, and hence one avoids knots.
908:
880:: on a manifold, codimension 1 is the dimension of topological disconnection by a submanifold, while codimension 2 is the dimension of
822:
335:
242:
151:
973:
833:
the number of constraints. We do not expect to be able to find a solution if the predicted codimension, i.e. the number of
17:
703:
value in this range. This statement is more perspicuous than the translation in terms of dimensions, because the
449:
749:, it is quite evident why dimensions add. The subspaces can be defined by the vanishing of a certain number of
74:
965:
949:
718:
996:
991:
944:
586:{\displaystyle \operatorname {codim} (W)=\dim(V/W)=\dim \operatorname {coker} (W\to V)=\dim(V)-\dim(W),}
881:
825:), and a constraint means we have to 'consume' a parameter to satisfy it, then the codimension of the
108:
another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector
1011:
616:
806:
789:
604:
1001:
939:
463:
of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition
603:
Finite-codimensional subspaces of infinite-dimensional spaces are often useful in the study of
754:
704:
78:
8:
785:
734:
82:
63:
1006:
877:
762:
124:
89:
969:
773:
750:
861:
722:
597:
430:
120:
43:
889:
865:
857:
853:
407:
70:
985:
415:
231:
957:
826:
434:
127:
85:. For this reason, the height of an ideal is often called its codimension.
47:
885:
846:
51:
31:
27:
Difference between the dimensions of mathematical object and a sub-object
746:
896:
815:
610:
460:
55:
39:
700:
615:
The fundamental property of codimension lies in its relation to
406:
Just as the dimension of a submanifold is the dimension of the
59:
596:
and is dual to the relative dimension as the dimension of the
860:
by methods of linear algebra, and for non-linear problems in
757:, their number is the codimension. Therefore, we see that
414:
the submanifold), the codimension is the dimension of the
396:{\displaystyle \operatorname {codim} (N)=\dim(M)-\dim(N).}
303:{\displaystyle \dim(W)+\operatorname {codim} (W)=\dim(V).}
212:{\displaystyle \operatorname {codim} (W)=\dim(V)-\dim(W).}
472:
338:
245:
154:
852:
The second is a matter of geometry, on the model of
796:
the two sets of constraints may not be independent;
799:the two sets of constraints may not be compatible.
784:In other language, which is basic for any kind of
585:
395:
302:
211:
788:, we are taking the union of a certain number of
983:
611:Additivity of codimension and dimension counting
841:(in the linear algebra case, there is always a
765:of the sets of linear functionals defining the
909:Glossary of differential geometry and topology
707:is just the sum of the codimensions. In words
803:The first of these is often expressed as the
856:; it is something that can be discussed for
448:is the dimension (possibly infinite) of the
410:(the number of dimensions that you can move
876:Codimension also has some clear meaning in
717:If the subspaces or submanifolds intersect
104:concept: it is only defined for one object
849:solution, which is therefore discounted).
772:. That union may introduce some degree of
145:is the difference between the dimensions:
792:. We have two phenomena to look out for:
222:It is the complement of the dimension of
871:
459:, which is more abstractly known as the
895:This quip is not vacuous: the study of
651:is their intersection with codimension
418:(the number of dimensions you can move
14:
984:
740:
956:
927:
433:of a (possibly infinite dimensional)
230:it adds up to the dimension of the
24:
317:is a submanifold or subvariety in
25:
1023:
226:in that, with the dimension of
921:
577:
571:
559:
553:
541:
535:
529:
511:
497:
485:
479:
387:
381:
369:
363:
351:
345:
294:
288:
276:
270:
258:
252:
203:
197:
185:
179:
167:
161:
75:projective algebraic varieties
13:
1:
966:Graduate Texts in Mathematics
914:
95:
77:, the codimension equals the
968:(Third ed.), Springer,
725:), codimensions add exactly.
7:
945:Encyclopedia of Mathematics
902:
10:
1028:
712:codimensions (at most) add
321:, then the codimension of
776:: the possible values of
761:is defined by taking the
753:, which if we take to be
729:This statement is called
605:topological vector spaces
818:to adjust (i.e. we have
440:then the codimension of
962:Advanced Linear Algebra
810:: if we have a number
805:principle of counting
587:
397:
304:
213:
872:In geometric topology
837:constraints, exceeds
588:
398:
305:
214:
42:idea that applies to
755:linearly independent
470:
336:
243:
152:
88:The dual concept is
786:intersection theory
741:Dual interpretation
735:intersection theory
731:dimension counting,
425:More generally, if
64:algebraic varieties
997:Geometric topology
992:Algebraic geometry
878:geometric topology
823:degrees of freedom
751:linear functionals
583:
422:the submanifold).
393:
300:
209:
125:finite-dimensional
90:relative dimension
18:Dimension counting
975:978-0-387-72828-5
774:linear dependence
100:Codimension is a
16:(Redirected from
1019:
1012:Dimension theory
978:
953:
931:
925:
862:projective space
745:In terms of the
733:particularly in
640:has codimension
626:has codimension
592:
590:
589:
584:
507:
402:
400:
399:
394:
309:
307:
306:
301:
218:
216:
215:
210:
81:of the defining
21:
1027:
1026:
1022:
1021:
1020:
1018:
1017:
1016:
982:
981:
976:
938:
935:
934:
926:
922:
917:
905:
874:
858:linear problems
771:
743:
690:
683:
672:
665:
646:
639:
632:
625:
613:
503:
471:
468:
467:
431:linear subspace
337:
334:
333:
244:
241:
240:
153:
150:
149:
121:linear subspace
98:
58:, and suitable
28:
23:
22:
15:
12:
11:
5:
1025:
1015:
1014:
1009:
1004:
1002:Linear algebra
999:
994:
980:
979:
974:
958:Roman, Stephen
954:
933:
932:
919:
918:
916:
913:
912:
911:
904:
901:
890:surgery theory
873:
870:
866:complex number
854:parallel lines
801:
800:
797:
769:
742:
739:
727:
726:
721:(which occurs
715:
693:
692:
688:
681:
670:
663:
644:
637:
630:
623:
612:
609:
594:
593:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
506:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
450:quotient space
408:tangent bundle
404:
403:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
313:Similarly, if
311:
310:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
220:
219:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
160:
157:
97:
94:
26:
9:
6:
4:
3:
2:
1024:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
989:
987:
977:
971:
967:
963:
959:
955:
951:
947:
946:
941:
940:"Codimension"
937:
936:
929:
924:
920:
910:
907:
906:
900:
898:
893:
891:
887:
883:
879:
869:
867:
863:
859:
855:
850:
848:
844:
840:
836:
832:
828:
824:
821:
817:
813:
809:
808:
798:
795:
794:
793:
791:
787:
782:
779:
775:
768:
764:
760:
756:
752:
748:
738:
736:
732:
724:
720:
719:transversally
716:
713:
710:
709:
708:
706:
702:
699:may take any
698:
687:
680:
676:
669:
662:
658:
657:
656:
654:
650:
643:
636:
629:
622:
618:
608:
606:
601:
599:
580:
574:
568:
565:
562:
556:
550:
547:
544:
538:
532:
526:
523:
520:
517:
514:
508:
504:
500:
494:
491:
488:
482:
476:
473:
466:
465:
464:
462:
458:
454:
451:
447:
443:
439:
436:
432:
428:
423:
421:
417:
416:normal bundle
413:
409:
390:
384:
378:
375:
372:
366:
360:
357:
354:
348:
342:
339:
332:
331:
330:
328:
324:
320:
316:
297:
291:
285:
282:
279:
273:
267:
264:
261:
255:
249:
246:
239:
238:
237:
236:
233:
232:ambient space
229:
225:
206:
200:
194:
191:
188:
182:
176:
173:
170:
164:
158:
155:
148:
147:
146:
144:
140:
136:
132:
129:
126:
122:
118:
113:
111:
107:
103:
93:
91:
86:
84:
80:
76:
72:
67:
65:
61:
57:
53:
49:
48:vector spaces
45:
41:
37:
33:
19:
961:
943:
923:
894:
882:ramification
875:
851:
842:
838:
834:
830:
827:solution set
819:
811:
804:
802:
783:
777:
766:
758:
744:
730:
728:
711:
696:
694:
685:
678:
674:
667:
660:
652:
648:
641:
634:
627:
620:
617:intersection
614:
602:
595:
456:
452:
445:
441:
437:
435:vector space
426:
424:
419:
411:
405:
326:
322:
318:
314:
312:
234:
227:
223:
221:
142:
138:
134:
130:
128:vector space
116:
114:
109:
105:
101:
99:
87:
68:
52:submanifolds
35:
29:
886:knot theory
864:, over the
847:null vector
835:independent
807:constraints
790:constraints
723:generically
135:codimension
133:, then the
38:is a basic
36:codimension
32:mathematics
986:Categories
930:, p. 93 §3
928:Roman 2008
915:References
897:embeddings
816:parameters
747:dual space
673:) ≤
647:, then if
96:Definition
1007:Dimension
950:EMS Press
569:
563:−
551:
536:→
527:
521:
495:
477:
379:
373:−
361:
343:
286:
268:
250:
195:
189:−
177:
159:
56:manifolds
44:subspaces
40:geometric
960:(2008),
903:See also
695:In fact
677:≤
655:we have
461:cokernel
102:relative
952:, 2001
868:field.
843:trivial
831:at most
701:integer
112:space.
60:subsets
972:
633:, and
598:kernel
106:inside
79:height
71:affine
763:union
659:max (
619:: if
524:coker
474:codim
429:is a
340:codim
265:codim
156:codim
123:of a
119:is a
83:ideal
50:, to
970:ISBN
884:and
73:and
69:For
829:is
814:of
705:RHS
566:dim
548:dim
518:dim
492:dim
444:in
420:off
376:dim
358:dim
329:is
325:in
283:dim
247:dim
192:dim
174:dim
141:in
137:of
115:If
110:sub
62:of
54:in
46:in
30:In
988::
964:,
948:,
942:,
845:,
737:.
684:+
666:,
607:.
600:.
412:on
235:V:
228:W,
224:W,
92:.
66:.
34:,
839:N
820:N
812:N
778:j
770:i
767:W
759:U
714:.
697:j
691:.
689:2
686:k
682:1
679:k
675:j
671:2
668:k
664:1
661:k
653:j
649:U
645:2
642:k
638:2
635:W
631:1
628:k
624:1
621:W
581:,
578:)
575:W
572:(
560:)
557:V
554:(
545:=
542:)
539:V
533:W
530:(
515:=
512:)
509:W
505:/
501:V
498:(
489:=
486:)
483:W
480:(
457:W
455:/
453:V
446:V
442:W
438:V
427:W
391:.
388:)
385:N
382:(
370:)
367:M
364:(
355:=
352:)
349:N
346:(
327:M
323:N
319:M
315:N
298:.
295:)
292:V
289:(
280:=
277:)
274:W
271:(
262:+
259:)
256:W
253:(
207:.
204:)
201:W
198:(
186:)
183:V
180:(
171:=
168:)
165:W
162:(
143:V
139:W
131:V
117:W
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.