Exotic sphere

An exotic sphere is a smooth manifold , that is homeomorphic but not diffeomorphic to the standard n - sphere .

Content

History

The first examples of exotic spheres were constructed by John Milnor in dimension 7; he proved that on $S^{7}$ ${\ displaystyle S ^ {7}}$ There are at least 7 different smooth structures. It is now known that on oriented $S^{7}$ ${\ displaystyle S ^ {7}}$ There are 28 different smooth structures (15 excluding orientation).

These examples, the so-called Milnor spheres , were found among spaces $S^{3}$ ${\ displaystyle S ^ {3}}$ - bundles over $S^{4}$ ${\ displaystyle S ^ {4}}$ . Such bundles are classified by two integers $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ - element $\mathbb {Z} ^{2}=\pi _{3}(\mathrm {SO} (4))$ ${\ displaystyle \ mathbb {Z} ^ {2} = \ pi _ {3} (\ mathrm {SO} (4))}$ . Some of these bundles $M_{a,b}$ ${\ displaystyle M_ {a, b}}$ are homeomorphic to the standard sphere, and at the same time are not diffeomorphic to it.

Insofar as $M_{a,b}$ ${\ displaystyle M_ {a, b}}$ simply connected, according to the generalized Poincaré conjecture , verification of homeomorphism $M_{a,b}$ ${\ displaystyle M_ {a, b}}$ and $S^{7}$ ${\ displaystyle S ^ {7}}$ comes down to homology counting $M_{a,b}$ ${\ displaystyle M_ {a, b}}$ ; this condition imposes certain conditions on $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ .

In the proof of non-diffeomorphism, Milnor argues on the contrary. He notes that diversity $M_{a,b}$ ${\ displaystyle M_ {a, b}}$ represent the boundary of an 8-dimensional manifold - space $W_{a,b}$ ${\ displaystyle W_ {a, b}}$ disk stratification $D^{4}$ ${\ displaystyle D ^ {4}}$ above $S^{4}$ ${\ displaystyle S ^ {4}}$ . Further, if $M_{a,b}$ ${\ displaystyle M_ {a, b}}$ diffeomorphic to the standard sphere, then $W_{a,b}$ ${\ displaystyle W_ {a, b}}$ can be glued with a ball to obtain a closed smooth 8-dimensional manifold. Counting the signature of the resulting manifold through its Pontryagin number leads to a contradiction.

Classification

The connected sum of two exotic n- dimensional spheres is also an exotic sphere. The connected sum operation turns various smooth structures on an oriented n- dimensional sphere into a monoid , called the monoid of exotic spheres .

n ≠ 4

For $n\neq 4$ ${\ displaystyle n \ neq 4}$ It is known that the monoid of exotic spheres is an abelian group called the group of exotic spheres .

This group is trivial for $n=1,2,3,5,6$ ${\ displaystyle n = 1,2,3,5,6}$ . That is, in these dimensions the existence of a homeomorphism onto the standard sphere $S^{n}$ ${\ displaystyle S ^ {n}}$ implies the existence of a diffeomorphism on $S^{n}$ ${\ displaystyle S ^ {n}}$ . At $n=7$ ${\ displaystyle n = 7}$ it is isomorphic to a cyclic group of order 28. That is, there exists a seven-dimensional exotic sphere $\Sigma ^{7}$ ${\ displaystyle \ Sigma ^ {7}}$ such that any 7-dimensional exotic sphere is diffeomorphic to the connected sum of several copies $\Sigma ^{7}$ ${\ displaystyle \ Sigma ^ {7}}$ ; with a connected amount of 28 copies $\Sigma ^{7}$ ${\ displaystyle \ Sigma ^ {7}}$ diffeomorphic to the standard sphere $S^{7}$ ${\ displaystyle S ^ {7}}$ .

The group of exotic spheres is isomorphic to the group Θ _{n of} classes of oriented h- cobordism of the homotopy n- sphere. This group is finite and abelian.

Group $\Theta _{n}$ ${\ displaystyle \ Theta _ {n}}$ has a cyclic subgroup

bP_{n+1}

{\ displaystyle bP_ {n + 1}}

,

appropriate $n$ ${\ displaystyle n}$ -spheres that restrict parallelizable manifolds .

If n is even, then the group $bP_{n+1}$ ${\ displaystyle bP_ {n + 1}}$ trivial
If a $n$ $\equiv$ $one$ $($ $mod$ $four$ $)$ {\ displaystyle n \ equiv 1 {\ pmod {4}}} then the group $b$ $P$ $n$ $+$ $one$ {\ displaystyle bP_ {n + 1}} has order 1 or 2
- It has an order of 1 for n = 1, 5, 13, 29, or 61.
- It has order 2 at $n\equiv 1{\pmod {4}}$ ${\ displaystyle n \ equiv 1 {\ pmod {4}}}$ if at the same time $n\neq 2^{k}-3$ ${\ displaystyle n \ neq 2 ^ {k} -3}$
If a $n$ $\equiv$ $3$ $($ $mod$ $four$ $)$ {\ displaystyle n \ equiv 3 {\ pmod {4}}} , i.e $n$ $+$ $one$ $=$ $four$ $m$ {\ displaystyle n + 1 = 4m} then at $m$ $\geq$ $2$ {\ displaystyle m \ geq 2} order is equal
- $|bP_{4m}|=2^{2m-2}(2^{2m-1}-1)B$ ${\ displaystyle | bP_ {4m} | = 2 ^ {2m-2} (2 ^ {2m-1} -1) B}$ ,

Where

B

{\ displaystyle B} Is a fraction numerator

|4B_{2m}/m|

{\ displaystyle | 4B_ {2m} / m |}

,

B_{2m}

{\ displaystyle B_ {2m}}

- Bernoulli numbers . (Sometimes the formula is slightly different due to different definitions of Bernoulli numbers.)

Factor groups $\Theta _{n}/bP_{n+1}$ ${\ displaystyle \ Theta _ {n} / bP_ {n + 1}}$ are described in terms of stable homotopy groups of spheres modulo the image of a J-homomorphism ). More precisely, there is an injective homomorphism

\Theta _{n}/bP_{n+1}\to \pi _{n}^{S}/J

{\ displaystyle \ Theta _ {n} / bP_ {n + 1} \ to \ pi _ {n} ^ {S} / J}

,

Where $\pi _{n}^{S}$ ${\ displaystyle \ pi _ {n} ^ {S}}$ Is the nth stable homotopy group of spheres, and $J$ ${\ displaystyle J}$ Is the image of a J -homomorphism. This homomorphism is either an isomorphism or has an image of index 2. The latter happens if and only if there exists an n- dimensional parallelizable manifold with Kerver invariant 1.

The question of the existence of such a manifold is called the Kerver problem. As of 2012, it is not resolved only for the case $n=126$ ${\ displaystyle n = 126}$ . Manifolds with Kerver 1 invariant were constructed in dimensions 2, 6, 14, 30, and 62.

Dimension n	one	2	3	four	five	6	7	eight	9	ten	eleven	12	13	14	15	sixteen	17	18	nineteen	20
Order Θ _n	one	one	one	one	one	one	28	2	eight	6	992	one	3	2	16256	2	sixteen	sixteen	523264	24
Order bP _{n +1}	one	one	one	one	one	one	28	one	2	one	992	one	one	one	8128	one	2	one	261632	one
Order Θ _n / bP _{n +1}	one	one	one	one	one	one	one	2	2 × 2	6	one	one	3	2	2	2	2 × 2 × 2	8 × 2	2	24
Order π _n ^S / J	one	2	one	one	one	2	one	2	2 × 2	6	one	one	3	2 × 2	2	2	2 × 2 × 2	8 × 2	2	24
Index	-	2	-	-	-	2	-	-	-	-	-	-	-	2	-	-	-	-	-	-

Further values in this table can be calculated from the information above together with a table of stable homotopy groups of spheres.

n = 4

In dimension $n=4$ ${\ displaystyle n = 4}$ practically nothing is known about the monoid of smooth spheres, except that it is finite or countably infinite and abelian. It is not known whether exotic smooth structures exist on a 4-dimensional sphere. The statement that they do not exist is known as the “Poincare smooth hypothesis”.

The so-called Glack twisting consists in cutting out a tubular neighborhood of the 2-sphere S ² in S ⁴ and gluing it back using a diffeomorphism of its boundary $S^{2}\times S^{1}$ ${\ displaystyle S ^ {2} \ times S ^ {1}}$ . The result is always homeomorphic to S ⁴ , but in most cases it is not known whether it is diffeomorphic to S ⁴ .

Twisted Spheres

Let a diffeomorphism be given $f\colon S^{n-1}\to S^{n-1}$ ${\ displaystyle f \ colon S ^ {n-1} \ to S ^ {n-1}}$ keeping orientation. Gluing two copies of the ball to display $f$ ${\ displaystyle f}$ between the boundaries, we get the so-called sphere crammed with diffeomorphism $f$ ${\ displaystyle f}$ . A twisted sphere is homeomorphic to the standard one, but, generally speaking, is not diffeomorphic to it.

In other words, a manifold is called a cramped sphere if it admits a Morse function with exactly two critical points.

For n ≠ 4, any exotic sphere is diffeomorphic to some twisted sphere.
For n = 4, any twisted sphere is diffeomorphic to the standard one.

Links

Akbulut, Selman (2009), Cappell – Shaneson homotopy spheres are standard , arXiv : 0907.0136
Brieskorn, Egbert V. (1966), "Examples of singular normal complex spaces which are topological manifolds", Proceedings of the National Academy of Sciences 55 (6): 1395–1397, doi : 10.1073 / pnas.55.6.1395 , MR 0198497 , PMC 224331 , PMID 16578636
Brieskorn, Egbert (1966b), "Beispiele zur Differentialtopologie von Singularitäten", Invent. Math. 2 (1): 1-14, doi : 10.1007 / BF01403388 , MR 0206972
Browder, William (1969), "The Kervaire invariant of framed manifolds and its generalization", Annals of Mathematics 90 (1): 157–186, doi : 10.2307 / 1970686 , JSTOR 1970686 , MR 0251736
Freedman, Michael; Gompf, Robert; Morrison, Scott; Walker, Kevin (2010), "Man and machine thinking about the smooth 4-dimensional Poincaré conjecture", Quantum Topology 1 (2): 171–208, arXiv : 0906.5177 , doi : 10.4171 / qt / 5
Gluck, Herman (1962), "The embedding of two-spheres in the four-sphere", Transactions of the American Mathematical Society 104 (2): 308–333, doi : 10.2307 / 1993581 , JSTOR 1993581 , MR 0146807
Hirzebruch, Friedrich; Mayer, Karl Heinz (1968), O (n) -Mannigfaligkeiten, Exotische Sphären und Singularitäten , Lecture Notes in Mathematics 57 , Berlin-New York: Springer-Verlag , doi : 10.1007 / BFb0074355 , MR 0229251 This book describes the works of Brieskorn, in whose exotic spheres are associated with the singularities of complex manifolds.
Kervaire, Michel A .; Milnor, John W. (1963). "Groups of homotopy spheres: I" (PDF). Annals of Mathematics (Princeton University Press) 77 (3): 504-537. doi : 10.2307 / 1970128 . JSTOR 1970128 . MR 0148075 . - This work describes the structure of a group of smooth structures on the n- sphere for n > 4. Unfortunately, the published article, “Groups of Homotopy Spheres: II,” never came out, but Levin’s lecture materials contain the material that she apparently could contain.
Levine, JP (1985), "Lectures on groups of homotopy spheres", Algebraic and geometric topology , Lecture Notes in Mathematics 1126 , Berlin-New York: Springer-Verlag, pp. 62–95, doi : 10.1007 / BFb0074439 , MR 8757031
Milnor, John W. (1956), "On manifolds homeomorphic to the 7-sphere", Annals of Mathematics 64 (2): 399–405, doi : 10.2307 / 1969983 , JSTOR 1969983 , MR 0082103
- J. Milnor. On varieties homeomorphic to a seven-dimensional sphere // Mathematics . - 1957. - No. 3 . - S. 35–42 .
Milnor, John W. (1959), "Sommes de variétes différentiables et structures différentiables des sphères" , Bulletin de la Société Mathématique de France 87 : 439–444, MR 0117744
Milnor, John W. (1959b), "Differentiable structures on spheres", American Journal of Mathematics 81 (4): 962–972, doi : 10.2307 / 2372998 , JSTOR 2372998 , MR 0110107
Milnor, John (2000), "Classification of ( n - 1) -connected 2 n -dimensional manifolds and the discovery of exotic spheres", in Cappell, Sylvain; Ranick, Andrew; Rosenberg, Jonathan, Surveys on Surgery Theory: Volume 1 , Annals of Mathematics Studies 145, Princeton University Press, pp. 25-30, ISBN 9780691049380, MR 1747528
Milnor, John Willard (2009), "Fifty years ago: topology of manifolds in the 50's and 60's", in Mrowka, Tomasz S .; Ozsváth., Peter S., Low dimensional topology. Lecture notes from the 15th Park City Mathematics Institute (PCMI) Graduate Summer School held in Park City, UT, Summer 2006. (PDF), IAS / Park City Math. Ser. 15 , Providence, RI: Amer. Math. Soc., Pp. 9–20, ISBN 978-0-8218-4766-4, MR 2503491
Milnor, John W. (2011), "Differential topology forty-six years later" (PDF), Notices of the American Mathematical Society 58 (6): 804–809
Rudyak, Yu.B. (2001), "Milnor sphere" (inaccessible link) , in Hazewinkel, Michiel, Encyclopedia of Mathematics , Springer, ISBN 978-1-55608-010-4

External links

Exotic Spheres on Manifold Atlas (link unavailable from 05/06/2016 [1198 days])
Exotic spheres. Source materials related to exotic areas.
Animations of factual exotic 7-spheres - video from a report by Niles Johnson from the Second Abel Conference .
Gluck_construction