1127:
39:. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are
552:-dimensional embeddings to embeddings of 2-disks. But this is not a reduction when the dimension is 4: the 2-disks themselves are middle-dimensional, so trying to embed them encounters exactly the same problems they are supposed to solve. This is the phenomenon that separates dimension 4 from others."
255:
to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial), there can be no algorithm to tell if two 4-manifolds have the same fundamental group. This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of
250:
it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. (More specifically, for any finitely presented group, one constructs a manifold with the given fundamental group, such that two manifolds in this family are homeomorphic if and only if the fundamental groups are
419:
to show that the smooth structures are different. Their results suggest that any classification of simply connected smooth 4-manifolds will be very complicated. There are currently no plausible conjectures about what this classification might look like. (Some early conjectures that all simply
399:
In contrast, very little is known about the second question of classifying the smooth structures on a smoothable 4-manifold; in fact, there is not a single smoothable 4-manifold where the answer is fully known. Donaldson showed that there are some simply connected compact 4-manifolds, such as
432:
There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in dimension 4. Here are some examples:
188:, there are two manifolds depending on the KirbyâSiebenmann invariant: one is 2-dimensional complex projective space, and the other is a fake projective space, with the same homotopy type but not homeomorphic (and with no smooth structure).
456:
In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countably-infinite number of non-diffeomorphic smooth
453:) vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing KirbyâSiebenmann invariant but no PL structure.
415:. Fintushel and Stern showed how to use surgery to construct large numbers of different smooth structures (indexed by arbitrary integral polynomials) on many different manifolds, using
195:
starts to increase extremely rapidly with the rank, so there are huge numbers of corresponding simply connected topological 4-manifolds (most of which seem to be of almost no interest).
127:
395:=7. See for recent (as of 2019) progress in this area.) The "11/8 conjecture" states that smooth structures do not exist if the dimension is less than 11/8 times the |signature|.
441:
provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its KirbyâSiebenmann invariant in H(
308:
If the form is indefinite and even we may as well assume that it is of nonpositive signature by changing orientations if necessary, in which case it is isomorphic to a sum of
275:
A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:
243:. If the fundamental group is too large (for example, a free group on 2 generators), then Freedman's techniques seem to fail and very little is known about such manifolds.
241:
219:
186:
494:
has been proved for all dimensions other than 4. In 4 dimensions, the PL Poincaré conjecture is equivalent to the smooth
Poincaré conjecture, and its truth is unknown.
501:
holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by
137:
and KirbyâSiebenmann invariant can arise, except that if the form is even, then the KirbyâSiebenmann invariant must be the signature/8 (mod 2).
286:
There is an almost complete answer to the first problem asking which simply connected compact 4-manifolds have smooth structures. First, the
523:
showed that there are topological manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.
720:
623:
653:; Lin, Jianfeng; Shi, XiaoLin; Xu, Zhouli (2019), "Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant",
404:, with a countably infinite number of different smooth structures. There are an uncountable number of different smooth structures on
340:(so that the dimension is at least 11/8 times the |signature|) then there is a smooth structure, given by taking a connected sum of
199:
Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is
545:
375:). This leaves a small gap between 10/8 and 11/8 where the answer is mostly unknown. (The smallest case not covered above has
1002:
383:=5, but this has also been ruled out, so the smallest lattice for which the answer is not currently known is the lattice II
581:
548:
uses an isotopy across an embedded 2-disk to simplify these intersections. Roughly speaking this reduces the study of
1152:
1110:
939:
859:
826:
885:
788:
1084:
97:
438:
130:
79:
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371:(so the dimension is at most 10/8 times the |signature|) then Furuta proved that no smooth structure exists (
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512:
decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
416:
1147:
1089:
919:
533:
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767:
Surgery and
Geometric Topology: Proceedings of a conference held at Josai University, Sakado, Sept. 1996
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287:
265:
192:
247:
224:
221:, there is a classification similar to the one above using Hermitian forms over the group ring of
202:
169:
846:, Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York,
302:) gives a complete answer: there is a smooth structure if and only if the form is diagonalizable.
764:
Quinn, F. (1996). "Problems in low-dimensional topology". In
Ranicki, A.; Yamasaki, M. (eds.).
505:). If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds.
295:
483:
145:
718:(2016). "Pin(2)-equivariant SeibergâWitten Floer homology and the Triangulation Conjecture".
1157:
1042:
Kirby, R. C.; Taylor, L. R. (1998). "A survey of 4-manifolds through the eyes of surgery".
1035:
1012:
971:
908:
869:
701:
636:
486:
is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see
268:
can be smoothed in an essentially unique way, so in particular the theory of 4 dimensional
28:
8:
814:
498:
421:
252:
1043:
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650:
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In the special case when the form is 0, this implies the 4-dimensional topological
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type of the manifold only depends on this intersection form, and on a
835:
473:
409:
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742:
948:
Furuta, Mikio (2001), "Monopole
Equation and the 11/8-Conjecture",
659:
985:, Lecture Notes in Mathematics, vol. 1374, Berlin, New York:
733:
282:
Classify the different smooth structures on a smoothable manifold.
272:
is much the same as the theory of 4 dimensional smooth manifolds.
1126:
527:
305:
If the form is indefinite and odd there is a smooth structure.
424:, possibly with orientations reversed, have been disproved.)
821:, Oxford Mathematical Monographs, Oxford: Clarendon Press,
472:
has an uncountable number of exotic smooth structures; see
508:
A topological manifold of dimension not equal to 4 has a
82:
on the middle dimensional homology. A famous theorem of
540:-dimensional submanifolds of a manifold of dimension 2
227:
205:
172:
100:
427:
191:
When the rank of the form is greater than about 28,
1105:, Providence, R.I.: American Mathematical Society,
809:
923:
256:many problems is already known to be intractable.
235:
213:
180:
121:
1139:
193:the number of positive definite unimodular forms
50:4-manifolds are important in physics because in
649:
834:
676:"Irrationality and the h-cobordism conjecture"
1018:
914:
616:"Differential topology forty-six years later"
1059:"Four-dimensional topology: an introduction"
881:"The topology of four-dimensional manifolds"
624:Notices of the American Mathematical Society
1041:
528:Failure of the Whitney trick in dimension 4
279:Which topological manifolds are smoothable?
1056:
264:For manifolds of dimension at most 6, any
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78:compact 4-manifold only depends on the
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294:If the intersection form is definite
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159:, a manifold not homeomorphic to any
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428:Special phenomena in 4 dimensions
1125:
886:Journal of Differential Geometry
789:Journal of Differential Geometry
437:In dimensions other than 4, the
1076:10.1090/S0273-0979-1980-14687-X
582:EnriquesâKodaira classification
490:). The Poincaré conjecture for
266:piecewise linear (PL) structure
1024:4-Manifolds and Kirby Calculus
1022:; Stipsicz, AndrĂĄs I. (1999),
819:The Geometry of Four-Manifolds
757:
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1103:The wild world of 4-manifolds
1028:American Mathematical Society
950:Mathematical Research Letters
844:Instantons and four-manifolds
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1085:"Four-dimensional manifolds"
674:Donaldson, Simon K. (1987).
236:{\displaystyle \mathbb {Z} }
214:{\displaystyle \mathbb {Z} }
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1090:Encyclopedia of Mathematics
983:The topology of 4-manifolds
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482:The solution to the smooth
460:Four is the only dimension
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932:Princeton University Press
439:KirbyâSiebenmann invariant
131:KirbyâSiebenmann invariant
963:10.4310/mrl.2001.v8.n3.a5
877:Freedman, Michael Hartley
852:10.1007/978-1-4684-0258-2
417:SeibergâWitten invariants
1153:Low-dimensional topology
1083:Matveev, S. V. (2001) ,
248:finitely presented group
1057:Mandelbaum, R. (1980),
926:Topology of 4-manifolds
66:Topological 4-manifolds
35:is a 4-manifold with a
1063:Bull. Amer. Math. Soc.
900:10.4310/jdg/1214437136
803:10.4310/jdg/1214437665
693:10.4310/jdg/1214441179
288:KirbyâSiebenmann class
251:isomorphic.) As there
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129:invariant called the
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1134:at Wikimedia Commons
1101:Scorpan, A. (2005),
916:Freedman, Michael H.
815:Kronheimer, Peter B.
680:J. Differential Geom
422:symplectic manifolds
351: − 3
225:
203:
170:
98:
84:Michael Freedman
29:topological manifold
930:, Princeton, N.J.:
840:Uhlenbeck, Karen K.
811:Donaldson, Simon K.
784:Donaldson, Simon K.
721:J. Amer. Math. Soc.
651:Hopkins, Michael J.
499:h-cobordism theorem
484:Poincaré conjecture
296:Donaldson's theorem
253:can be no algorithm
151:If the form is the
146:Poincaré conjecture
90:) implies that the
27:is a 4-dimensional
1148:Geometric topology
995:10.1007/BFb0089031
773:. pp. 97â104.
716:Manolescu, Ciprian
517:simplicial complex
402:Dolgachev surfaces
260:Smooth 4-manifolds
233:
211:
178:
161:simplicial complex
119:
52:General Relativity
16:Mathematical space
1130:Media related to
1004:978-3-540-51148-9
567:Algebraic surface
521:Ciprian Manolescu
80:intersection form
60:pseudo-Riemannian
33:smooth 4-manifold
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1020:Gompf, Robert E.
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743:10.1090/jams829
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546:"Whitney trick"
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324:(â1) for some
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488:exotic sphere
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312:copies of II
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270:PL manifolds
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62:4-manifold.
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41:homeomorphic
32:
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1158:4-manifolds
1132:4-manifolds
956:: 279â291,
727:: 147â176.
534:Frank Quinn
497:The smooth
457:structures.
373:Furuta 2001
345:K3 surfaces
320:copies of E
157:E8 manifold
21:mathematics
1142:Categories
660:1812.04052
598:References
577:5-manifold
572:3-manifold
510:handlebody
464:for which
355:copies of
153:E8 lattice
140:Examples:
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556:See also
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