769:
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.
580:
877:. 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
390:
297:
206:
888:
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.
458:
881:
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.
897:
869:: on a manifold, codimension 1 is the dimension of topological disconnection by a submanifold, while codimension 2 is the dimension of
811:
324:
231:
140:
962:
822:
the number of constraints. We do not expect to be able to find a solution if the predicted codimension, i.e. the number of
692:
value in this range. This statement is more perspicuous than the translation in terms of dimensions, because the
438:
738:, it is quite evident why dimensions add. The subspaces can be defined by the vanishing of a certain number of
63:
954:
938:
707:
985:
980:
933:
575:{\displaystyle \operatorname {codim} (W)=\dim(V/W)=\dim \operatorname {coker} (W\to V)=\dim(V)-\dim(W),}
870:
814:), and a constraint means we have to 'consume' a parameter to satisfy it, then the codimension of the
97:
another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector
1000:
605:
795:
778:
593:
990:
928:
452:
of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition
592:
Finite-codimensional subspaces of infinite-dimensional spaces are often useful in the study of
743:
693:
67:
8:
774:
723:
71:
52:
995:
866:
751:
113:
78:
958:
762:
739:
850:
711:
586:
419:
109:
32:
878:
854:
846:
842:
396:
59:
974:
404:
220:
946:
815:
423:
116:
74:. For this reason, the height of an ideal is often called its codimension.
36:
874:
835:
40:
20:
16:
Difference between the dimensions of mathematical object and a sub-object
735:
885:
804:
599:
449:
44:
28:
689:
604:
The fundamental property of codimension lies in its relation to
395:
Just as the dimension of a submanifold is the dimension of the
48:
585:
and is dual to the relative dimension as the dimension of the
849:
by methods of linear algebra, and for non-linear problems in
746:, their number is the codimension. Therefore, we see that
403:
the submanifold), the codimension is the dimension of the
385:{\displaystyle \operatorname {codim} (N)=\dim(M)-\dim(N).}
292:{\displaystyle \dim(W)+\operatorname {codim} (W)=\dim(V).}
201:{\displaystyle \operatorname {codim} (W)=\dim(V)-\dim(W).}
461:
327:
234:
143:
841:
The second is a matter of geometry, on the model of
785:
the two sets of constraints may not be independent;
788:the two sets of constraints may not be compatible.
773:In other language, which is basic for any kind of
574:
384:
291:
200:
777:, we are taking the union of a certain number of
972:
600:Additivity of codimension and dimension counting
830:(in the linear algebra case, there is always a
754:of the sets of linear functionals defining the
898:Glossary of differential geometry and topology
696:is just the sum of the codimensions. In words
792:The first of these is often expressed as the
845:; it is something that can be discussed for
437:is the dimension (possibly infinite) of the
399:(the number of dimensions that you can move
865:Codimension also has some clear meaning in
706:If the subspaces or submanifolds intersect
93:concept: it is only defined for one object
838:solution, which is therefore discounted).
761:. That union may introduce some degree of
134:is the difference between the dimensions:
781:. We have two phenomena to look out for:
211:It is the complement of the dimension of
860:
448:, which is more abstractly known as the
884:This quip is not vacuous: the study of
640:is their intersection with codimension
407:(the number of dimensions you can move
973:
729:
945:
916:
422:of a (possibly infinite dimensional)
219:it adds up to the dimension of the
13:
306:is a submanifold or subvariety in
14:
1012:
215:in that, with the dimension of
910:
566:
560:
548:
542:
530:
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518:
500:
486:
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265:
259:
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192:
186:
174:
168:
156:
150:
64:projective algebraic varieties
1:
955:Graduate Texts in Mathematics
903:
84:
66:, the codimension equals the
957:(Third ed.), Springer,
714:), codimensions add exactly.
7:
934:Encyclopedia of Mathematics
891:
10:
1017:
701:codimensions (at most) add
310:, then the codimension of
765:: the possible values of
750:is defined by taking the
742:, which if we take to be
718:This statement is called
594:topological vector spaces
807:to adjust (i.e. we have
429:then the codimension of
951:Advanced Linear Algebra
799:: if we have a number
794:principle of counting
576:
386:
293:
202:
861:In geometric topology
826:constraints, exceeds
577:
387:
294:
203:
31:idea that applies to
744:linearly independent
459:
325:
232:
141:
77:The dual concept is
775:intersection theory
730:Dual interpretation
724:intersection theory
720:dimension counting,
414:More generally, if
53:algebraic varieties
986:Geometric topology
981:Algebraic geometry
867:geometric topology
812:degrees of freedom
740:linear functionals
572:
411:the submanifold).
382:
289:
198:
114:finite-dimensional
79:relative dimension
964:978-0-387-72828-5
763:linear dependence
89:Codimension is a
1008:
1001:Dimension theory
967:
942:
920:
914:
851:projective space
734:In terms of the
722:particularly in
629:has codimension
615:has codimension
581:
579:
578:
573:
496:
391:
389:
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383:
298:
296:
295:
290:
207:
205:
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70:of the defining
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1007:
1006:
1005:
971:
970:
965:
927:
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863:
847:linear problems
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672:
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420:linear subspace
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110:linear subspace
87:
47:, and suitable
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12:
11:
5:
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991:Linear algebra
988:
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947:Roman, Stephen
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907:
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901:
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893:
890:
879:surgery theory
862:
859:
855:complex number
843:parallel lines
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789:
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728:
716:
715:
710:(which occurs
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439:quotient space
397:tangent bundle
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369:
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302:Similarly, if
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15:
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3:
2:
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989:
987:
984:
982:
979:
978:
976:
966:
960:
956:
952:
948:
944:
940:
936:
935:
930:
929:"Codimension"
926:
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913:
909:
899:
896:
895:
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880:
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872:
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844:
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764:
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749:
745:
741:
737:
727:
725:
721:
713:
709:
708:transversally
705:
702:
699:
698:
697:
695:
691:
688:may take any
687:
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669:
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651:
647:
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645:
643:
639:
632:
625:
618:
611:
607:
597:
595:
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588:
569:
563:
557:
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533:
527:
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512:
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480:
477:
471:
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462:
455:
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453:
451:
447:
443:
440:
436:
432:
428:
425:
421:
417:
412:
410:
406:
405:normal bundle
402:
398:
379:
373:
367:
364:
361:
355:
349:
346:
343:
337:
331:
328:
321:
320:
319:
317:
313:
309:
305:
286:
280:
274:
271:
268:
262:
256:
253:
250:
244:
238:
235:
228:
227:
226:
225:
222:
221:ambient space
218:
214:
195:
189:
183:
180:
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171:
165:
162:
159:
153:
147:
144:
137:
136:
135:
133:
129:
125:
121:
118:
115:
111:
107:
102:
100:
96:
92:
82:
80:
75:
73:
69:
65:
61:
56:
54:
50:
46:
42:
38:
37:vector spaces
34:
30:
26:
22:
950:
932:
912:
883:
871:ramification
864:
840:
831:
827:
823:
819:
816:solution set
808:
800:
793:
791:
772:
766:
755:
747:
733:
719:
717:
700:
685:
683:
674:
667:
663:
656:
649:
641:
637:
630:
623:
616:
609:
606:intersection
603:
591:
584:
445:
441:
434:
430:
426:
424:vector space
415:
413:
408:
400:
394:
315:
311:
307:
303:
301:
223:
216:
212:
210:
131:
127:
123:
119:
117:vector space
105:
103:
98:
94:
90:
88:
76:
57:
41:submanifolds
24:
18:
875:knot theory
853:, over the
836:null vector
824:independent
796:constraints
779:constraints
712:generically
124:codimension
122:, then the
27:is a basic
25:codimension
21:mathematics
975:Categories
919:, p. 93 §3
917:Roman 2008
904:References
886:embeddings
805:parameters
736:dual space
662:) ≤
636:, then if
85:Definition
996:Dimension
939:EMS Press
558:
552:−
540:
525:→
516:
510:
484:
466:
368:
362:−
350:
332:
275:
257:
239:
184:
178:−
166:
148:
45:manifolds
33:subspaces
29:geometric
949:(2008),
892:See also
684:In fact
666:≤
644:we have
450:cokernel
91:relative
941:, 2001
857:field.
832:trivial
820:at most
690:integer
101:space.
49:subsets
961:
622:, and
587:kernel
95:inside
68:height
60:affine
752:union
648:max (
608:: if
513:coker
463:codim
418:is a
329:codim
254:codim
145:codim
112:of a
108:is a
72:ideal
39:, to
959:ISBN
873:and
62:and
58:For
818:is
803:of
694:RHS
555:dim
537:dim
507:dim
481:dim
433:in
409:off
365:dim
347:dim
318:is
314:in
272:dim
236:dim
181:dim
163:dim
130:in
126:of
104:If
99:sub
51:of
43:in
35:in
19:In
977::
953:,
937:,
931:,
834:,
726:.
673:+
655:,
596:.
589:.
401:on
224:V:
217:W,
213:W,
81:.
55:.
23:,
828:N
809:N
801:N
767:j
759:i
756:W
748:U
703:.
686:j
680:.
678:2
675:k
671:1
668:k
664:j
660:2
657:k
653:1
650:k
642:j
638:U
634:2
631:k
627:2
624:W
620:1
617:k
613:1
610:W
570:,
567:)
564:W
561:(
549:)
546:V
543:(
534:=
531:)
528:V
522:W
519:(
504:=
501:)
498:W
494:/
490:V
487:(
478:=
475:)
472:W
469:(
446:W
444:/
442:V
435:V
431:W
427:V
416:W
380:.
377:)
374:N
371:(
359:)
356:M
353:(
344:=
341:)
338:N
335:(
316:M
312:N
308:M
304:N
287:.
284:)
281:V
278:(
269:=
266:)
263:W
260:(
251:+
248:)
245:W
242:(
196:.
193:)
190:W
187:(
175:)
172:V
169:(
160:=
157:)
154:W
151:(
132:V
128:W
120:V
106:W
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