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It is known that this list is strictly increasing in difficulty in the sense that there are examples of contact 3-manifolds with weak but no strong filling, and others that have strong but no Stein filling. Further, it can be shown that each type of filling is an example of the one preceding it, so
547:, but it has been shown that any semi-filling can be modified to be a filling of the same type, of the same 3-manifold, in the symplectic world (Stein manifolds always have one boundary component).
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There are many types of fillings, and a few examples of these types (within a probably limited perspective) follow.
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that a Stein filling is a strong symplectic filling, for example. It used to be that one spoke of
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H. Geiges, An
Introduction to Contact Topology, Cambridge University Press, 2008
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All the following cobordisms are oriented, with the orientation on
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at each point of the boundary is the one pointing directly out of
76: = 3, where one may consider certain types of fillings.
72:. Perhaps the most active area of current research is when
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that are complex with respect to the complex structure on
103:, which is the one where the first basis vector of the
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455:{\displaystyle \{x\in \mathbb {C} ^{2}:|x|=1\}}
8:
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526:| < 1} bounded by that sphere.
115:. Mathematicians call this orientation the
252:symplectic filling of a contact manifold (
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99:is given by the boundary orientation of
396:. The canonical example of this is the
357:A Stein filling of a contact manifold (
16:Cobordism W between X and the empty set
558:A Few Remarks about Symplectic Filling
127:given by a symplectic structure. Let
68: + 1)-dimensional manifold
7:
384:is the set of complex tangencies to
238:{\displaystyle \omega |_{\xi }>0}
388:– that is, those tangent planes to
87:filling of any orientable manifold
535:in this context, which means that
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14:
484:{\displaystyle \mathbb {C} ^{2}}
462:where the complex structure on
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378:strictly pseudoconvex boundary
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1:
95:such that the orientation of
511:{\displaystyle {\sqrt {-1}}}
347:{\displaystyle \partial W=X}
305:near the boundary (which is
290:{\displaystyle \partial W=X}
260:) is a symplectic manifold (
200:{\displaystyle \partial W=X}
309:) and α is a primitive for
111:, with respect to a chosen
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49:. More to the point, the
564:, 2004, p. 277–293
560:, Geometry and Topology
539:is one of possibly many
518:in each coordinate and
582:(2004), p. 73–80
576:On Symplectic Fillings
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578:Algebr. Geom. Topol.
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491:is multiplication by
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117:outward normal first
91:is another manifold
55:topological manifold
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168:symplectic manifold
602:Geometric topology
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113:Riemannian metric
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156:contact manifold
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367:Stein manifold
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53:-dimensional
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570:math/0311459
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137:contact form
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574:J. Etnyre,
313:. That is,
131:denote the
119:convention.
21:mathematics
551:References
372:which has
297:such that
207:such that
152:symplectic
501:−
413:∈
333:∂
276:∂
225:ξ
215:ω
186:∂
47:empty set
36:cobordism
596:Category
398:3-sphere
85:oriented
62:boundary
45:and the
41:between
29:manifold
376:as its
365:) is a
268:) with
178:) with
166:) is a
135:of the
64:of an (
60:is the
25:filling
584:online
250:strong
139:
133:kernel
566:arXiv
321:in a
303:exact
34:is a
27:of a
380:and
230:>
149:weak
23:, a
543:of
301:is
83:An
19:In
598::
319:dα
317:=
248:A
147:A
143:.
580:4
568::
562:8
545:W
537:X
524:x
520:W
504:1
477:2
472:C
450:}
447:1
444:=
440:|
436:x
432:|
428::
423:2
418:C
410:x
407:{
394:W
390:X
386:X
382:ξ
374:X
370:W
363:ξ
361:,
359:X
354:.
342:X
339:=
336:W
315:ω
311:ω
307:X
299:ω
285:X
282:=
279:W
266:ω
264:,
262:W
258:ξ
256:,
254:X
245:.
233:0
220:|
195:X
192:=
189:W
176:ω
174:,
172:W
170:(
164:ξ
162:,
160:X
158:(
141:α
129:ξ
125:W
109:W
101:W
97:X
93:W
89:X
74:n
70:W
66:n
58:X
51:n
43:X
39:W
32:X
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