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In [[differential topology]], an '''exotic sphere''' is a [[differentiable manifold]] ''M'' that is [[homeomorphic]] but not [[diffeomorphic]] to the standard Euclidean ''n''-[[sphere]]. That is, ''M'' is a sphere from the point of view of all its topological properties, but carrying a [[smooth structure]] that is not the familiar one (hence the name "exotic").
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The first exotic spheres were constructed by {{harvs|authorlink=John Milnor|first=John|last=Milnor|year=1956|txt=yes}} in  dimension ''n'' = 7 as ''S''<sup>3</sup>-[[Fiber bundle|bundles]] over ''S''<sup>4</sup>. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension {{harvtxt|Milnor|1959}} showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian [[monoid]] under connected sum, which is a [[Finite group|finite]] [[abelian group]] if the dimension is not 4. The classification of exotic spheres by {{harvs |authorlink1=Michel Kervaire |first1=Michel |last1=Kervaire |first2=John |last2=Milnor |year=1963 |txt=yes}} showed that the [[orientability|oriented]] exotic 7-spheres are the non-trivial elements of a [[cyclic group]] of order 28 under the operation of [[connected sum]].
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==Introduction==
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The unit ''n''-sphere, ''S''<sup>n</sup>, is the set of all [[tuple|''n''+1-tuples]] (''x''<sub>1</sub>, ''x''<sub>2</sub>, ... ''x''<sub>n+1</sub>) of real numbers, such that the sum ''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> + ... + ''x''<sub>n+1</sub><sup>2</sup> = 1. (''S''<sup>1</sup> is a circle; ''S''<sup>2</sup> is the surface of an ordinary ball of radius one in 3 dimensions.) Topologists consider a space, ''X'', to be an ''n''-sphere if every point in ''X'' can be assigned to exactly one point in the unit ''n''-sphere in a [[continuous function|continuous]] way, which means that sufficiently nearby points in ''X'' get assigned to nearby points in ''S''<sup>n</sup> and vice-versa. For example a point ''x'' on an ''n''-sphere of radius ''r'' can be matched with a point on the unit ''n''-sphere by adjusting its distance from the origin by 1/''r''. 


In [[differential topology]], a more stringent condition is added, that the functions matching points in ''X'' with points in ''S''<sup>n</sup> should be [[smooth function|smooth]], that is they should have [[derivative]]s of all orders everywhere. To calculate derivatives, one needs to have local coordinate systems defined consistently in ''X.'' Mathematicians were surprised in 1956 when John Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6- or 12-spheres. Some higher dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an important unsolved problem in mathematics.
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==Classification==
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The monoid of [[smooth structure]]s on ''n''-spheres is the collection of oriented smooth ''n''-manifolds which are homeomorphic to the ''n''-sphere, taken up to orientation-preserving diffeomorphism.  The monoid operation is the [[connected sum]]. Provided ''n'' ≠ 4,  this monoid is a group and is isomorphic to the group Θ<sub>''n''</sub>  of [[H-cobordism|''h''-cobordism]] classes of oriented [[homotopy sphere|homotopy ''n''-spheres]], which is finite and abelian. In dimension 4 almost nothing is known about the monoid of smooth spheres, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite; see the section on [[Exotic_sphere#4-dimensional exotic spheres and Gluck twists|Gluck twist]]s. All homotopy ''n''-spheres are homeomorphic to the ''n''-sphere by the generalized [[Poincaré conjecture]], proved by [[Stephen Smale]] in dimensions bigger than 4, [[Michael Freedman]] in dimension 4, and [[Grigori Perelman]] in dimension 3. In dimension 3, [[Edwin E. Moise]] proved that every topological manifold has an essentially unique smooth structure (see [[Moise's theorem]]), so the monoid of smooth structures on the 3-sphere is trivial.


=== Parallelizable manifolds===
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The group Θ<sub>''n''</sub> has a cyclic subgroup
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:<math>bP_{n+1}\ </math>
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represented by ''n''-spheres that bound [[parallelizable manifold]]s. The structures of ''bP''<sub>''n''+1</sub> and the quotient
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:<math>\Theta_n/bP_{n+1}\ </math>
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The group ''bP''<sub>''n''+1</sub> is a cyclic group, and is trivial or order 2 except in case <math>n = 4k+3,</math> in which case it can be large, with its order related to the [[Bernoulli number]]s. It is trivial if ''n'' is even. If ''n'' is 1 mod 4 it has order 1 or 2; in particular it has order 1 if ''n'' is 1, 5,  13,  29, or 61, and {{harvtxt|Browder|1969}} proved that it has order 2 if ''n'' = 1 mod 4 is not of the form 2<sup>''k''</sup> – 3.
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The order of ''bP''<sub>4''n''</sub> for ''n''&nbsp;≥&nbsp;2 is


:<math>2^{2n-2}(2^{2n-1}-1)B \,\!</math>
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where ''B'' is the numerator of |4''B''<sub>2''n''</sub>/''n''|, and ''B''<sub>2''n''</sub> is a [[Bernoulli number]]. (The formula in the topological literature differs slightly because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)
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===Map between quotients===
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The quotient group Θ<sub>''n''</sub>/''bP''<sub>''n''+1</sub> has a description in terms of [[stable homotopy groups of spheres]] modulo the image of the [[J-homomorphism]]); it is either equal to the quotient or index 2. More precisely there is an injective map
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:<math>\Theta_n/bP_{n+1}\to \pi_n^S/J\,</math>
<ul>
where π<sub>''n''</sub><sup>''S''</sup> is the ''n''th stable homotopy group of spheres, and ''J'' is the image of the ''J''-homomorphism. As with ''bP''<sub>''n''+1</sub>, the image of ''J'' is a cyclic group, and is trivial or order 2 except in case <math>n = 4k+3,</math> in which case it can be large, with its order related to the [[Bernoulli number]]s. The quotient group <math>\pi_n^S/J\,</math> is the "hard" part of the stable homotopy groups of spheres, and accordingly <math>\Theta_n/bP_{n+1}</math> is the hard part of the exotic spheres, but almost completely reduces to computing homotopy groups of spheres. The map is either an isomorphism (the image is the whole group), or an injective map with [[Index of a subgroup|index]] 2. The latter is the case if and only if there exists an ''n''-dimensional framed manifold with [[Kervaire invariant]] 1, which is known as the [[Kervaire invariant problem]]. Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem.
 
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{{as of|2012}}, the Kervaire invariant problem is almost completely solved, with only the case ''n''&nbsp;=&nbsp;126 remaining open; see that article for details. This is primarily the work of {{harvtxt|Browder|1969}}, which proved that such manifolds only existed in dimension ''n''&nbsp;=&nbsp;2<sup>''j''</sup>&nbsp;&minus;&nbsp;2, and {{harvtxt|Hill|Hopkins|Ravenel|2009}}, which proved that there were no such manifolds for dimension <math>254 = 2^8 - 2</math> and above. Manifolds with Kervaire invariant 1 have been constructed in dimension 2, 6, 14, 30, and 62, but dimension 126 is open, with no manifold being either constructed or disproven.
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===Order of Θ<sub>''n''</sub>===
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The order of the group Θ<sub>''n''</sub> is given in this table {{OEIS|id=A001676}} from {{harv|Kervaire|Milnor|1963}} (except that the entry for ''n'' = 19 is wrong by a factor of 2 in their paper; see the correction in volume III p.&nbsp;97 of Milnor's collected works).
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:{| border="1" cellpadding="2" class="wikitable" style="text-align:center"
== New Balance Sale  "Thrown Out" ==
|-
! Dim  n !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20
|-
! order Θ<sub>''n''</sub>
| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24
|-
!''bP''<sub>''n''+1</sub>
| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 1 || 2 || 1 || 992 || 1 || 1 || 1 || 8128 || 1 || 2 || 1 || 261632 || 1
|-
!Θ<sub>''n''</sub>/''bP''<sub>''n''+1</sub>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||2&times;2 || 6 || 1 || 1 || 3 || 2 || 2 || 2 || 2&times;2&times;2 || 8&times;2 || 2 || 24
|-
!π<sub>''n''</sub><sup>''S''</sup>/''J''
| 1 || 2 || 1 || 1 || 1 || 2 || 1 || 2 || 2&times;2 || 6 || 1 || 1 || 3 || 2&times;2 || 2 || 2 || 2&times;2&times;2 || 8&times;2 || 2 || 24
|-
!index
| - || 2 || - || - || - || 2 || - || - || - || - || - || - || - || 2 || - || - || - || - || - || -
|}


Further entries in this table can be computed from the information above together with the table of [[stable homotopy groups of spheres]].
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==Explicit examples of exotic spheres==
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|quote=  When I came upon such an example in the mid-50s, I was very puzzled and didn’t know what to make of it. At first, I thought I’d found a counterexample to the generalized Poincaré conjecture in dimension seven. But careful study showed that the manifold really was homeomorphic to S<sup>7</sup>. Thus, there exists a differentiable structure on S<sup>7</sup> not diffeomorphic to the standard one.
 
|source={{harvs|txt|first=John|last=Milnor|year=2009|loc=p.12}}
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One of the first examples of an exotic sphere found by {{harvtxt|Milnor|1956|loc=section 3}} was the following: Take two copies of [[Ball (mathematics)|''B''<sup>4</sup>]]&times;''S''<sup>3</sup>, each with [[Boundary (topology)|boundary]] ''S''<sup>3</sup>&times;''S''<sup>3</sup>, and glue them together by identifying (''a'',''b'') in the boundary with (''a'', ''a''<sup>2</sup>''ba''<sup>&minus;1</sup>), (where we identify each ''S''<sup>3</sup> with the group of unit [[quaternion]]s). The resulting manifold has a natural smooth structure and is homeomorphic to ''S''<sup>7</sup>, but is not diffeomorphic to ''S''<sup>7</sup>. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a [[Morse function]] with just two [[critical point (mathematics)|critical points]], both non-degenerate, which implies that it is topologically a sphere.
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As shown by {{harvs|txt=yes|first=Egbert |last=Brieskorn|year=1966|year2=1966b}} (see also {{harv|Hirzebruch|Mayer|1968}}) the intersection of the [[complex manifold]] of points in '''C'''<sup>5</sup> satisfying
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:<math>a^2 + b^2 + c^2 + d^3 + e^{6k-1} = 0\ </math>
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with a small sphere around the origin for ''k'' = 1, 2, ..., 28 gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called [[Brieskorn sphere]]s.
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==Twisted spheres==
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Given an (orientation-preserving) diffeomorphism ''f'':&nbsp;''S''<sup>''n''−1</sup>→''S''<sup>''n''−1</sup>, gluing the boundaries of two copies of the standard disk ''D''<sup>''n''</sup> together by <math>f</math> yields a manifold called a ''twisted sphere'' (with ''twist'' ''f''). It is homotopy equivalent to the standard ''n-''sphere because the gluing map is homotopic to the identity (being an orientation-preserving diffeomorphism, hence degree 1), but not in general diffeomorphic to the standard sphere. {{harv|Milnor|1959b}}
Setting <math>\Gamma_n</math> to be the group of twisted ''n-''spheres (under connect sum), one obtains the exact sequence
:<math>\pi_0\,\text{Diff}^+(D^n) \to \pi_0\,\text{Diff}^+(S^{n-1}) \to \Gamma_n \to 0. \,\!</math>


For ''n''&nbsp;&gt;&nbsp;4, every exotic sphere is diffeomorphic to a twisted sphere, a result proven by [[Stephen Smale]]. (In contrast, in the [[Piecewise linear manifold|piecewise linear]] setting the left-most map is onto via [[Alexander Trick#Radial extension|radial extension]]: there are no piecewise-linear-twisted spheres.) The group Γ<sub>''n''</sub> of twisted spheres is always isomorphic
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to the group Θ<sub>''n''</sub>. The notations are different because it was not known at first that they were the same for ''n''=3 or 4; for example, the case ''n''=3 is equivalent to the [[Poincaré conjecture]].
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In 1970 Jean Cerf proved the [[pseudoisotopy theorem]] which implies that <math>\pi_0\,\text{Diff}^+(D^n)</math> is the trivial group provided <math>n \geq 6</math>, so <math>\Gamma_n \simeq \pi_0\,\text{Diff}^+(S^{n-1})</math> provided <math>n \geq 6</math>.
 
 
  <li>[http://www.468bet.com/thread-2737134-1-1.html http://www.468bet.com/thread-2737134-1-1.html]</li>
==Applications==
 
If ''M'' is a [[piecewise linear manifold]] then the problem of finding the compatible smooth structures on ''M'' depends on knowledge of the groups Γ<sub>''k''</sub> = Θ<sub>''k''</sub>. More precisely, the obstructions to the existence of any smooth structure lie in the groups H<sub>''k+1''</sub>(''M'', Γ<sub>''k''</sub>) for various values of ''k'', while if such a smooth structure exists then all such smooth structures can be  classified using the groups H<sub>''k''</sub>(''M'', Γ<sub>''k''</sub>).
  <li>[http://www.hmssxcz.com/Review.asp?NewsID=564 http://www.hmssxcz.com/Review.asp?NewsID=564]</li>
In particular the groups Γ<sub>''k''</sub> vanish if ''k''&lt;7, so all PL manifolds of dimension at most 7 have a smooth structure, which is essentially unique if the manifold has dimension at most 6.
 
 
  <li>[http://bbs.0312mc.com/forum.php?mod=viewthread&tid=263498 http://bbs.0312mc.com/forum.php?mod=viewthread&tid=263498]</li>
The following finite abelian groups are essentially the same:
 
*The group Θ<sub>''n''</sub> of h-cobordism classes of oriented homotopy ''n''-spheres.
  <li>[http://lab.hijike.com/forum.php?mod=viewthread&tid=2163454&fromuid=704069 http://lab.hijike.com/forum.php?mod=viewthread&tid=2163454&fromuid=704069]</li>
*The group  of h-cobordism classes of oriented ''n''-spheres.
 
*The group  Γ<sub>''n''</sub> of twisted oriented ''n'' spheres.
  <li>[http://www.chinaiprlaw.com/bbs/forum.php?mod=viewthread&tid=258177 http://www.chinaiprlaw.com/bbs/forum.php?mod=viewthread&tid=258177]</li>
*The homotopy group π<sub>''n''</sup>(PL/DIFF)
 
*If ''n'' ≠ 3, the homotopy π<sub>''n''</sup>(TOP/DIFF) (if ''n''=3 this group has order 2; see [[Kirby–Siebenmann invariant]]).
</ul>
*The group of smooth structures of an oriented PL ''n''-sphere.
*If ''n''≠4, the group of smooth structures of an oriented topological ''n''-sphere.
*If ''n''&ne;5, the group of components of the group of all orientation-preserving diffeomorphisms of ''S''<sup>''n''&minus;1</sup>.
 
==4-dimensional exotic spheres and Gluck twists==
In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincaré conjecture", and is discussed by {{harvs|txt | last1=Freedman | first1=Michael | last2=Gompf | first2=Robert | last3=Morrison | first3=Scott | last4=Walker | first4=Kevin | year=2010}} who say that it is believed to be false.
 
Some candidates for exotic 4-spheres are given by '''Gluck twists''' {{harv|Gluck|1962}}. These are constructed by cutting out a tubular neighborhood of a 2-sphere ''S'' in ''S''<sup>4</sup> and gluing it back in using a diffeomorphism of its boundary ''S''<sup>2</sup>&times;''S''<sup>1</sup>. The result is always homeomorphic to ''S''<sup>4</sup>. But in most cases it is unknown whether or not the result is diffeomorphic to  ''S''<sup>4</sup>. (If the 2-sphere is unknotted, or given by spinning a knot in the 3-sphere, then the Gluck twist is known to be diffeomorphic to ''S''<sup>4</sup>, but there are plenty of other ways to knot a 2-sphere in ''S''<sup>4</sup>.)
 
{{harvtxt|Akbulut|2009}} showed that a certain family of candidates for 4-dimensional exotic spheres constructed by  Cappell and Shaneson  are in fact standard.
 
==See also==
*[[Atlas (topology)]]
*[[Clutching construction]]
*[[Exotic R4|Exotic '''R'''<sup>4</sup>]]
*[[Cerf theory]]
*[[Seven-dimensional space]]
 
==References==
*{{Citation | last1=Akbulut | first1=Selman | title=Cappell–Shaneson homotopy spheres are standard | arxiv=0907.0136 | year=2009}}
*{{citation |mr=0198497 |last=Brieskorn |first= Egbert V. |title=Examples of singular normal complex spaces which are topological manifolds |journal=[[Proceedings of the National Academy of Sciences]] |volume= 55|year= 1966|pages= 1395–1397 |issue=6 |doi=10.1073/pnas.55.6.1395}}
*{{citation |mr=0206972 |last=Brieskorn |first= Egbert |title=Beispiele zur Differentialtopologie von Singularitäten |journal=Invent. Math. |volume=2 |year=1966b |issue=1 |pages=1–14 |doi=10.1007/BF01403388}}
*{{citation |mr=0251736 |first=William |last=Browder |title=The Kervaire invariant of framed manifolds and its generalization |journal= [[Annals of Mathematics]] |volume= 90 |year=1969 |pages= 157–186 |doi=10.2307/1970686 |jstor=1970686 |issue=1}}
*{{Citation | last1=Freedman | first1=Michael | last2=Gompf | first2=Robert | last3=Morrison | first3=Scott | last4=Walker | first4=Kevin | title=Man and machine thinking about the smooth 4-dimensional Poincaré conjecture | arxiv=0906.5177 | year=2010 | journal= Quantum Topology | volume=1 | issue=2 | pages=171–208}}
*{{citation |first=Herman |last=Gluck |title=The embedding of two-spheres in the four-sphere |journal=[[Transactions of the American Mathematical Society]] |volume=104 |year=1962 |pages=308&ndash;333 |mr= 0146807 |issue=2 |doi=10.2307/1993581 |jstor=1993581 }}
*{{citation |first=Friedrich |last=Hirzebruch |first2= Karl Heinz |last2=Mayer |title=O(n)-Mannigfaligkeiten, Exotische Sphären und Singularitäten |series= Lecture Notes in Mathematics |volume= 57 |publisher= [[Springer-Verlag]] |publication-place= Berlin-New York |year= 1968 |mr= 0229251 |doi=10.1007/BFb0074355}} This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds.
*{{cite jstor|1970128}} – This paper describes the structure of the group of smooth structures on an ''n''-sphere for ''n'' > 4. Sadly, the promised paper "Groups of Homotopy Spheres: II" never appeared, but Levine's lecture notes contain the material which it might have been expected to contain.
*{{citation |first=J.P. |last=Levine |title=Algebraic and geometric topology |chapter=Lectures on groups of homotopy spheres |series=Lecture Notes in Mathematics |volume=1126 |publisher=Springer-Verlag |publication-place=Berlin-New York |year=1985 |pages=62&ndash;95 |mr=8757031 |doi=10.1007/BFb0074439}}
*{{citation|first=John W.|last= Milnor |title=On manifolds homeomorphic to the 7-sphere|journal= Annals of Mathematics |volume=64|year=1956|issue= 2 |pages=399&ndash;405|mr= 0082103|doi=10.2307/1969983|jstor=1969983}}
*{{citation|first=John W.|last= Milnor|url=http://www.numdam.org/item?id=BSMF_1959__87__439_0 |title=Sommes de variétes différentiables et structures différentiables des sphères|journal=Bulletin de la Société Mathématique de France |volume=87|year=1959|pages= 439&ndash;444|mr= 0117744}}
*{{citation|first=John W.|last= Milnor |title=Differentiable structures on spheres|journal= [[American Journal of Mathematics]]|volume=81|year=1959b|issue= 4|pages= 962&ndash;972|mr= 0110107|doi=10.2307/2372998|jstor=2372998}}
*{{citation|mr=1747528|last= Milnor|first= John |chapter=Classification of (''n''&nbsp;&minus;&nbsp;1)-connected 2''n''-dimensional manifolds and the discovery of exotic spheres|title=Surveys on surgery theory|pages=25–30| title = Surveys on Surgery Theory: Volume 1 | series = Annals of Mathematics Studies 145 | editor1-first = Sylvain | editor1-last = Cappell | editor2-first = Andrew | editor2-last = Ranick | editor3-first = Jonathan | editor3-last = Rosenberg|year= 2000 | publisher = Princeton University Press | isbn = 9780691049380}}.
*{{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | editor1-last=Mrowka | editor1-first=Tomasz S. | editor2-last=Ozsváth. | editor2-first=Peter S. | title=Low dimensional topology. Lecture notes from the 15th Park City Mathematics Institute (PCMI) Graduate Summer School held in Park City, UT, Summer 2006. | url=http://www.math.sunysb.edu/~jack/PREPRINTS/pcity-lec.pdf | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=IAS/Park City Math. Ser. | isbn=978-0-8218-4766-4  | mr=2503491 | year=2009 | volume=15 | chapter=Fifty years ago: topology of manifolds in the 50's and 60's | pages=9–20}}
*{{citation|first=John W.|last= Milnor |title=Differential topology forty-six years later|journal= [[Notices of the American Mathematical Society]] |volume=58|year=2011|issue= 6 |pages=804&ndash;809|url=http://www.ams.org/notices/201106/rtx110600804p.pdf}}
*{{springer|title=Milnor sphere|id=M/m063800|first=Yu.B.| last=Rudyak}}
 
== External links ==
*[http://www.manifoldatlas.him.uni-bonn.de/Exotic_spheres Exotic spheres] on the Manifold Atlas
*[http://www.maths.ed.ac.uk/~aar/exotic.htm Exotic sphere home page] on the home page of Andrew Ranicki. Assorted source material relating to exotic spheres.
{{Use dmy dates|date=April 2011}}
*[http://www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres] Video from a presenation by [http://www.nilesjohnson.net/ Niles Johnson] at the [http://www.ima.umn.edu/2011-2012/SW1.30-2.1.12/ Second Abel conference] in honor of [[John Milnor]].
*[http://www.map.mpim-bonn.mpg.de/Gluck_construction#References The Gluck construction] on the Manifold Atlas
{{DEFAULTSORT:Exotic Sphere}}
[[Category:Differential topology]]
[[Category:Differential structures]]
[[Category:Surgery theory]]

Latest revision as of 11:16, 12 November 2014

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