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| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. She is really fond of caving but she doesn't have the time lately. For a while I've been in Alaska but I will have to move in a year or two. He is an information officer.<br><br>Look into my web blog: telephone psychic ([http://www.theempowerblog.com/gwpblog/well-show-you-all-sorts-of-things-about-hobbies/ Full Document]) |
| {{DISPLAYTITLE:''h''-Cobordism}}
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| In [[geometric topology]] and [[differential topology]], an (''n''+1)-dimensional [[cobordism]] ''W'' between ''n''-dimensional [[manifold]]s ''M'' and ''N'' is an '''''h''-cobordism''' (the ''h'' stands for [[homotopy equivalence]]) if the inclusion maps
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| : <math> M \hookrightarrow W \quad\mbox{and}\quad N \hookrightarrow W</math>
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| are homotopy equivalences.
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| The '''''h''-cobordism theorem''' gives sufficient conditions for an ''h''-cobordism to be trivial, i.e., to be '''Cat'''-isomorphic to the cylinder ''M'' × [0, 1]. Here '''Cat''' refers to any of the categories of '''[[smooth manifold|smooth]]''', '''[[piecewise linear manifold|piecewise linear]]''', or '''[[topological manifold|topological]]''' manifolds.
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| The theorem was first proved by [[Stephen Smale]] for which he received the [[Fields Medal]] and is the fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the [[Generalized Poincaré Conjecture]].
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| ==Background==
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| Before Smale proved this theorem, mathematicians had got stuck trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The ''h''-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "[[Whitney embedding theorem|Whitney trick]]" of [[Hassler Whitney]], which geometrically untangles homologically-tangled spheres of complementary dimension in a manifold of dimension >5. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that [[4-manifold#Failure of the Whitney trick in dimension 4|the trick fails to work]] in lower dimensions, which have no room for untanglement.
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| ==Precise statement of the ''h''-cobordism theorem==
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| Let ''n'' be at least 5 and let ''W'' be a compact (''n''+1)-dimensional ''h''-cobordism between ''M'' and ''N'' in the category '''Cat'''='''[[smooth manifold|Diff]]''', '''[[piecewise linear manifold|PL]]''', or '''[[topological manifold|Top]]''' such that ''W'', ''M'' and ''N'' are [[simply connected]], then ''W'' is '''Cat'''-isomorphic to ''M'' × [0, 1]. The isomorphism can be chosen to be the identity on ''M'' × {0}.
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| This means that the homotopy equivalence between M, W, and N is homotopic to a '''Cat'''-isomorphism.
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| ==Low dimensions==
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| For ''n'' = 4, the ''h''-cobordism theorem is true topologically (proved by [[Michael Freedman]] using a 4-dimensional Whitney trick) but is false PL and smoothly (as shown by [[Simon Donaldson]]). | |
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| For ''n'' = 3, the ''h''-cobordism theorem for smooth manifolds has not been proved and, due to the [[Poincaré conjecture]], is equivalent to the hard open question of whether the 4-sphere has non-standard [[smooth structure]]s.
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| For ''n'' = 2, the ''h''-cobordism theorem<ref>In 3 dimensions and below, the categories are the same: '''Diff''' = '''PL''' = '''Top'''.</ref> is true – it is equivalent to the [[Poincaré conjecture]], which has been proved by [[Grigori Perelman]].
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| For ''n'' = 1, ''h''-cobordism theorem is vacuously true, since there is no closed simply-connected 1-dimensional manifold.
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| For ''n'' = 0, the ''h''-cobordism theorem is trivially true: the interval is the only connected cobordism between connected 0-manifolds.
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| ==A proof sketch==
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| A [[Cobordism#Morse functions|Morse function]] <math>f:W\to[a,b]</math> induces a [[handle decomposition]] of ''W'', i.e., if there is a single critical point of index ''k'' in <math>f^{-1}([c,c'])</math>, then the ascending cobordism <math>W_{c'}</math> is obtained from <math>W_c</math> by attaching a ''k''-handle. The goal of the proof is to find a handle decomposition with no handles at all so that integrating the non-zero gradient vector field of ''f'' gives the desired diffeomorphism to the trivial cobordism.
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| This is achieved through a series of techniques.
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| '''1) Handle Rearrangement'''
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| First, we want to rearrange all handles by order so that lower order handles are attached first. The question is thus when can we slide an ''i''-handle off of a ''j''-handle? This can be done by a radial isotopy so long as the ''i'' attaching sphere and the ''j'' belt sphere do not intersect. We thus want <math>(i-1)+(n-j)\leq\dim\partial W-1=n-1</math> which is equivalent to <math>i\leq j</math>.
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| We then define the handle chain complex <math>(C_*,\partial_*)</math> by letting <math>C_k</math> be the free abelian group on the ''k''-handles and defining <math>\partial_k:C_k\to C_{k-1}</math> by sending a ''k''-handle <math>h_{\alpha}^k</math> to <math>\sum_{\beta}<h_{\alpha}^k|h_{\beta}^{k-1}>h_{\beta}^{k-1}</math>, where <math><h_{\alpha}^k|h_{\beta}^{k-1}></math> is the intersection number of the ''k''-attaching sphere and the (''k''-1)-belt sphere.
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| '''2) Handle Cancellation'''
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| Next, we want to "cancel" handles. The idea is that attaching a ''k''-handle <math>h_{\alpha}^k</math> might create a hole that can be filled in by attaching a (''k''+1)-handle <math>h_{\beta}^{k+1}</math>. This would imply that <math>\partial_{k+1}h_{\beta}^{k+1}=\pm h_{\alpha}^k</math> and so the <math>(\alpha,\beta)</math> entry in the matrix of <math>\partial_{k+1}</math> would be <math>\pm 1</math>. However, when is this condition sufficient? That is, when can we geometrically cancel handels if this condition is true? The answer lies in carefully analyzing when the manifold remains simply-connected after removing the attaching and belt spheres in question, and finding an embedded disk using the [[Whitney embedding theorem|Whitney trick]]. This analysis leads to the requirement that ''n'' must be at least 5. Moreover, during the proof one requires that the cobordism has no 0-,1-,''n''-, or ''(n+1)''-handles which is obtained by the next technique.
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| '''3) Handle Trading'''
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| The idea of handle trading is to create a cancelling pair of (''k''+1)- and (''k''+2)-handles so that a given ''k''-handle cancels with the (''k''+1)-handle leaving behind the (''k''+2)-handle. To do this, consider the core of the ''k''-handle which is an element in <math>\pi_k(W,M)</math>. This group is trivial since ''W'' is an ''h''-cobordism. Thus, there is a disk <math>D^{k+1}</math> which we can fatten to a cancelling pair as desired, so long as we can embed this disk into the boundary of ''W''. This embedding exists if <math>\dim\partial W-1=n-1\geq 2(k+1)</math>. Since we are assuming ''n'' is at least 5 this means that ''k'' is either 0 or 1. Finally, by considering the negative of the given Morse function, ''-f'', we can turn the handle decomposition upside down and also remove the ''n''- and (''n''+1)-handles as desired.
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| '''4) Handle Sliding'''
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| Finally, we want to make sure that doing row and column operations on <math>\partial_k</math> corresponds to a geometric operation. Indeed, it isn't hard to show (best done by drawing a picture) that sliding a ''k''-handle <math>h_{\alpha}^k</math> over another ''k''-handle <math>h_{\beta}^k</math> replaces <math>h_{\alpha}^k</math> by <math>h_{\alpha}^k\pm h_{\beta}^k</math> in the basis for <math>C_k</math>.
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| The proof of the theorem now follows: the handle chain complex is exact since <math>H_*(W,M;\mathbb{Z})=0</math>. Thus <math>C_k\cong \mathrm{coker}\,\partial_{k+1}\oplus\mathrm{im}\,\partial_{k+1}</math> since the <math>C_k</math> are free. Then <math>\partial_k</math>, which is an integer matrix, restricts to an invertible morphism which can thus be diagonalized via elementary row operations (handle sliding) and must have only <math>\pm 1</math> on the diagonal because it is invertible. Thus, all handles are paired with a single other cancelling handle yielding a decomposition with no handles.
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| ==The ''s''-cobordism theorem ==
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| If the assumption that ''M'' and ''N'' are simply connected is dropped, ''h''-cobordisms need not be cylinders; the obstruction is exactly the [[Whitehead torsion]] τ (''W'', ''M'') of the inclusion <math>M \hookrightarrow W</math>.
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| Precisely, the '''''s''-cobordism theorem''' (the ''s'' stands for [[simple-homotopy equivalence]]), proved independently by [[Barry Mazur]], [[John Stallings]], and [[Dennis Barden]], states (assumptions as above but where ''M'' and ''N'' need not be simply connected):
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| : An ''h''-cobordism is a cylinder if and only if [[Whitehead torsion]] τ (''W'', ''M'') vanishes. | |
| The torsion vanishes if and only if the inclusion <math>M \hookrightarrow W</math> is not just a homotopy equivalence, but a [[simple homotopy equivalence]].
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| Note that one need not assume that the other inclusion <math>N \hookrightarrow W</math> is also a simple homotopy equivalence—that follows from the theorem.
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| Categorically, ''h''-cobordisms form a [[groupoid]].
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| Then a finer statement of the ''s''-cobordism theorem is that the isomorphism classes of this groupoid (up to '''Cat'''-isomorphism of ''h''-cobordisms) are [[torsor]]s for the respective<ref>Note that identifying the Whitehead groups of the various manifolds requires that one choose base points <math>m\in M, n\in N</math> and a path in ''W'' connecting them.</ref> [[Whitehead group]]s Wh(π), where <math>\pi \cong \pi_1(M) \cong \pi_1(W) \cong \pi_1(N).</math>
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| ==Notes==
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| <references/>
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| == See also ==
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| * [[Semi-s-cobordism|Semi-''s''-cobordism]]
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| ==References==
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| * Freedman, Michael H.; Quinn, Frank, ''Topology of 4-manifolds'', Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. viii+259 pp. ISBN 0-691-08577-3. This does the theorem for topological 4-manifolds.
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| *[[John Milnor|Milnor, John]], ''Lectures on the h-cobordism theorem'', notes by L. Siebenmann and J. Sondow, [[Princeton University Press]], Princeton, NJ, 1965. v+116 pp. This gives the proof for smooth manifolds.
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| *Rourke, Colin Patrick; Sanderson, Brian Joseph, ''Introduction to piecewise-linear topology'', Springer Study Edition, [[Springer-Verlag]], Berlin-New York, 1982. ISBN 3-540-11102-6. This proves the theorem for PL manifolds.
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| *S. Smale, "On the structure of manifolds" Amer. J. Math., 84 (1962) pp. 387–399
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| *{{springer|id=H/h046010|title=h-cobordism|first=Yu.B.|last= Rudyak}}
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| {{DEFAULTSORT:H-Cobordism}}
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| [[Category:Differential topology]]
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| [[Category:Surgery theory]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. She is really fond of caving but she doesn't have the time lately. For a while I've been in Alaska but I will have to move in a year or two. He is an information officer.
Look into my web blog: telephone psychic (Full Document)