22:
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168:" indicates what kind of manifold can be decomposed; the column labeled "How it is decomposed" indicates how, starting with a manifold, one can decompose it into smaller pieces; the column labeled "The pieces" indicates what the pieces can be; and the column labeled "How they are combined" indicates how the smaller pieces are combined to make the large manifold.
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Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a
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gives that every 3-manifold has a unique triangulation, unique up to common subdivision. In dimension 4, not all manifolds are triangulable. For higher dimensions, general existence of triangulations is unknown.
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as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form
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The table below is a summary of the various manifold-decomposition techniques. The column labeled "
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the manifold, then cut along special surfaces (condition on boundary curves and sutures...)
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373:
8:
496:
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340:
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253:
410:
366:
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290:
519:
478:
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345:
545:
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424:
390:
328:
234:
554:
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413:. Note that the handles must generally be added in a specific order.
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manifold. In particular, it has been useful in attempts to classify
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510:
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union by the trivial homeomorphism along the resultant boundaries
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310:. The decomposition is unique when the manifold is orientable.
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Union along their boundary, using the trivial homeomorphism
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Unions along subsurfaces on boundaries of handlebodies
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Depends on dimension. In dimension 3, a theorem by
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429:Cut along a sequence of incompressible surfaces
367:Union along the boundary by some homeomorphism
248:Glue together pairs of codimension-one faces
157:and also in proving the higher-dimensional
50:. Unsourced material may be challenged and
254:Jaco-Shalen/Johannson torus decomposition
114:Learn how and when to remove this message
386:(and the decomposition is never unique)
553:
411:Union along a subset of the boundaries
141:may be decomposed or split by writing
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48:adding citations to reliable sources
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14:
572:
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20:
393:a handle is associated to each
530:Three orientable handlebodies
1:
7:
539:
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63:"Manifold decomposition"
473:Open book decomposition
223:How they are combined
209:Type of decomposition
217:How it is decomposed
374:Handle decomposition
188:adding missing items
44:improve this article
441:Disk decomposition
313:Cut along embedded
298:Prime decomposition
274:Cut along embedded
159:Poincaré conjecture
561:Geometric topology
341:Heegaard splitting
186:; you can help by
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495:and a family of
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503:with that link
419:Haken hierarchy
391:Morse functions
363:of equal genus
329:Prime manifolds
285:Seifert-fibered
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546:Surgery theory
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425:Haken manifold
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395:critical point
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321:with disjoint
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235:Edwin E. Moise
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130:, a branch of
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499:that share a
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378:Any compact (
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334:Connected sum
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229:Triangulation
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182:This list is
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104:December 2009
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65: –
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59:Find sources:
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39:
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34:
29:This article
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23:
18:
17:
522:3-manifolds
361:handlebodies
302:Essentially
287:3-manifolds
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101:
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42:Please help
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497:2-manifolds
453:3-manifolds
353:3-manifolds
308:3-manifolds
270:3-manifolds
259:Irreducible
220:The pieces
195:August 2008
155:3-manifolds
132:mathematics
485:3-manifold
482:orientable
450:orientable
384:n-manifold
350:orientable
263:orientable
184:incomplete
74:newspapers
526:Surgeries
281:Atoroidal
244:Simplices
31:does not
555:Category
540:See also
511:Trigenus
501:boundary
444:Certain
403:(called
389:Through
304:surfaces
136:manifold
128:topology
516:Compact
464:3-balls
446:compact
433:3-balls
405:handles
317:; then
315:spheres
267:compact
88:scholar
52:removed
37:sources
520:closed
479:closed
458:Suture
380:smooth
346:Closed
90:
83:
76:
69:
61:
401:Balls
323:balls
95:JSTOR
81:books
493:link
477:Any
423:Any
359:Two
306:and
276:tori
134:, a
67:news
35:any
33:cite
283:or
190:.
126:In
46:by
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