Unimodular lattice: Difference between revisions
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{{Short description|Integral lattice of determinant 1 or −1}} |
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{{distinguish|modular lattice}} |
{{distinguish|modular lattice}} |
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In [[geometry]] and mathematical [[group theory]], a '''unimodular lattice''' is an integral [[Lattice (group)|lattice]] of [[Lattice (group)#Dividing space according to a lattice|determinant]] |
In [[geometry]] and mathematical [[group theory]], a '''unimodular lattice''' is an integral [[Lattice (group)|lattice]] of [[Lattice (group)#Dividing space according to a lattice|determinant]] 1 or −1. For a lattice in ''n''-dimensional [[Euclidean space]], this is equivalent to requiring that the [[volume]] of any [[fundamental domain]] for the lattice be 1. |
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The [[E8 lattice|''E''<sub>8</sub> lattice]] and the [[Leech lattice]] are two famous examples. |
The [[E8 lattice|''E''<sub>8</sub> lattice]] and the [[Leech lattice]] are two famous examples. |
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== Definitions == |
== Definitions == |
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* A '''lattice''' is a [[free abelian group]] of finite [[ |
* A '''lattice''' is a [[free abelian group]] of finite [[free abelian group|rank]] with a [[symmetric bilinear form]] (·, ·). |
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* The lattice is '''integral''' if (·,·) takes integer values. |
* The lattice is '''integral''' if (·,·) takes [[integer]] values. |
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* The '''dimension''' of a lattice is the same as its rank (as a '''Z'''-[[module (mathematics)|module]]). |
* The '''dimension''' of a lattice is the same as its [[free module|rank]] (as a '''Z'''-[[module (mathematics)|module]]). |
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* The '''norm''' of a lattice element ''a'' is (''a'', ''a''). |
* The '''norm''' of a lattice element ''a'' is (''a'', ''a''). |
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* A lattice is '''positive definite''' if the norm of all nonzero elements is positive. |
* A lattice is '''positive definite''' if the norm of all nonzero elements is positive. |
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* The '''determinant''' of a lattice is the determinant of the [[Gram matrix]], a matrix with entries '' |
* The '''determinant''' of a lattice is the [[determinant]] of the [[Gram matrix]], a [[matrix (mathematics)|matrix]] with entries (''a<sub>i</sub>'', ''a<sub>j</sub>''), where the elements ''a<sub>i</sub>'' form a basis for the lattice. |
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* An integral lattice is '''unimodular''' if its determinant is 1 or −1. |
* An integral lattice is '''unimodular''' if its determinant is 1 or −1. |
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* A unimodular lattice is '''even''' or '''type II''' if all norms are even, otherwise '''odd''' or '''type I'''. |
* A unimodular lattice is '''even''' or '''type II''' if all norms are even, otherwise '''odd''' or '''type I'''. |
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* The '''minimum''' of a positive definite lattice is the lowest nonzero norm. |
* The '''minimum''' of a positive definite lattice is the lowest nonzero norm. |
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* Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is '''positive definite''', '''Lorentzian''', and so on if its vector space is. |
* Lattices are often embedded in a [[real number|real]] [[vector space]] with a symmetric bilinear form. The lattice is '''positive definite''', '''Lorentzian''', and so on if its vector space is. |
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* The '''signature''' of a lattice is the [[signature (quadratic form)|signature]] of the form on the vector space. |
* The '''signature''' of a lattice is the [[signature (quadratic form)|signature]] of the form on the vector space. |
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== Properties == |
== Properties == |
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An integral lattice is unimodular [[if and only if]] its [[dual lattice]] is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual. |
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Given a pair (''m'',''n'') of nonnegative integers, an even unimodular lattice of signature (''m'',''n'') exists if and only if ''m |
Given a pair (''m'',''n'') of nonnegative integers, an even unimodular lattice of signature (''m'',''n'') exists if and only if ''m''−''n'' is divisible by 8, but an odd unimodular lattice of signature (''m'',''n'') always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8. Examples in all admissible signatures are given by the ''II<sub>m,n</sub>'' and ''I<sub>m,n</sub>'' constructions, respectively. |
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The [[theta function]] of a unimodular positive definite lattice is a [[modular form]] whose weight is one half the rank. If the lattice is even, the form has [[level of a modular form|level]] 1, and if the lattice is odd the form has Γ<sub>0</sub>(4) structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣''n''/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80,<ref>{{cite web |url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/unimodular.html |title=Unimodular Lattices, Together With A Table of the Best Such Lattices |last1=Nebe |first1=Gabriele |last2=Sloane |first2=Neil |website=Online Catalogue of Lattices | |
The [[theta function]] of a unimodular positive definite lattice is a [[modular form]] whose weight is one half the rank. If the lattice is even, the form has [[level of a modular form|level]] 1, and if the lattice is odd the form has Γ<sub>0</sub>(4) structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣''n''/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80,<ref>{{cite web |url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/unimodular.html |title=Unimodular Lattices, Together With A Table of the Best Such Lattices |last1=Nebe |first1=Gabriele |last2=Sloane |first2=Neil |website=Online Catalogue of Lattices |access-date=2015-05-30}}</ref> and their non-existence has been [[mathematical proof|proven]] for dimensions above 163,264.<ref>{{cite book |last=Nebe |first=Gabriele |author-link=Gabriele Nebe |arxiv=1201.1834 |chapter=Boris Venkov's Theory of Lattices and Spherical Designs |year=2013 | title=Diophantine methods, lattices, and arithmetic theory of quadratic forms | pages=1–19 | publisher=[[American Mathematical Society]] | location=Providence, RI | mr=3074799 | volume=587 | series=Contemporary Mathematics | editor1-last=Wan | editor1-first=Wai Kiu | editor2-last=Fukshansky | editor2-first=Lenny | editor3-last=Schulze-Pillot | editor3-first=Rainer | editor4-last=Vaaler | editor4-first=Jeffrey D.| display-editors = 3|bibcode=2012arXiv1201.1834N }}</ref> |
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==Classification == |
==Classification == |
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For indefinite lattices, the classification is easy to describe. |
For indefinite lattices, the classification is easy to describe. |
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Write ''R''<sup>m,n</sup> for the ''m+n'' dimensional vector space |
Write '''R'''<sup>''m'',''n''</sup> for the ''m'' + ''n'' [[dimension (vector space)|dimensional]] vector space '''R'''<sup>''m''+''n''</sup> with the inner product of |
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''R''<sup>m+n</sup> with the inner product of |
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: <math> a_1 b_1 + \cdots + a_m b_m - a_{m+1} b_{m+1} - \cdots - a_{m+n} b_{m+n}. \, </math> |
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:''a''<sub>1</sub>''b''<sub>1</sub>+...+''a''<sub>m</sub>''b''<sub>m</sub> − ''a''<sub>m+1</sub>''b''<sub>m+1</sub> − ... − ''a''<sub>m+n</sub>''b''<sub>m+n</sub>. |
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In ''R''<sup>m,n</sup> there is one odd indefinite unimodular lattice up to isomorphism, |
In '''R'''<sup>''m'',''n''</sup> there is one odd indefinite unimodular lattice up to [[isomorphism]], |
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denoted by |
denoted by |
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:''I<sub>m,n</sub> |
:''I''<sub>''m'',''n''</sub>, |
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which is given by all vectors (''a''<sub>1</sub>,...,''a''<sub>m+n</sub>) |
which is given by all vectors (''a''<sub>1</sub>,...,''a''<sub>''m''+''n''</sub>) |
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in ''R''<sup>m,n</sup> with all the ''a<sub>i</sub> |
in '''R'''<sup>''m'',''n''</sup> with all the ''a''<sub>''i''</sub> integers. |
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There are no indefinite even unimodular lattices unless |
There are no indefinite even unimodular lattices unless |
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in which case there is a unique example up to isomorphism, denoted by |
in which case there is a unique example up to isomorphism, denoted by |
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:''II<sub>m,n</sub> |
:''II''<sub>''m'',''n''</sub>. |
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This is given by all vectors (''a''<sub>1</sub>,...,''a''<sub>m+n</sub>) |
This is given by all vectors (''a''<sub>1</sub>,...,''a''<sub>''m''+''n''</sub>) |
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in ''R''<sup>m,n</sup> such that either all the ''a<sub>i</sub>'' are integers or they are all integers |
in '''R'''<sup>''m'',''n''</sup> such that either all the ''a<sub>i</sub>'' are integers or they are all integers plus 1/2, and their sum is even. The lattice ''II''<sub>8,0</sub> is the same as the ''E''<sub>8</sub> lattice. |
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plus 1/2, and their sum is even. |
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The lattice ''II<sub>8,0</sub>'' is the same as the ''E<sub>8</sub>'' lattice. |
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Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example ''I<sub>n,0</sub> |
Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example ''I''<sub>''n'',0</sub> in each dimension ''n'' |
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less than 8, and two examples (''I<sub>8,0</sub> |
less than 8, and two examples (''I''<sub>8,0</sub> and ''II''<sub>8,0</sub>) in dimension 8. The number of lattices increases moderately up to dimension 25 (where there |
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are 665 of them), but beyond dimension 25 the [[Smith-Minkowski-Siegel mass formula]] implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 in dimension 32. |
are 665 of them), but beyond dimension 25 the [[Smith-Minkowski-Siegel mass formula]] implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 in dimension 32. |
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In some sense unimodular lattices up to dimension 9 are controlled by ''E''<sub>8</sub>, and up to dimension 25 they are controlled by the Leech lattice, and this accounts for their unusually good behavior in these dimensions. For example, the [[Dynkin diagram]] of the norm-2 vectors of unimodular lattices in dimension up to 25 can be naturally identified with a configuration of vectors in the Leech lattice. The wild increase in numbers beyond 25 dimensions might be attributed to the fact that these lattices are no longer controlled by the Leech lattice. |
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In some sense unimodular lattices up to dimension 9 are controlled by |
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''E<sub>8</sub>'', and up to dimension 25 they are controlled by the |
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Leech lattice, and this accounts for their unusually good behavior |
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in these dimensions. For example, the [[Dynkin diagram]] of the norm |
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2 vectors of unimodular lattices in dimension up to 25 can be naturally |
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identified with a configuration of vectors in the Leech lattice. The wild increase in numbers beyond 25 dimensions might be attributed to the fact that these lattices are no longer controlled by |
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the Leech lattice. |
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Even positive definite unimodular lattice exist only in dimensions divisible by 8. |
Even positive definite unimodular lattice exist only in dimensions divisible by 8. |
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There is one in dimension 8 (the ''E''<sub>8</sub> lattice), two in dimension 16 (''E''<sub>8</sub><sup>2</sup> and ''II''<sub>16,0</sub>), and 24 in dimension 24, called the [[Niemeier lattice]]s (examples: the [[Leech lattice]], ''II''<sub>24,0</sub>, ''II''<sub>16,0</sub> + ''II''<sub>8,0</sub>, ''II''<sub>8,0</sub><sup>3</sup>). Beyond 24 dimensions the number increases very rapidly; in 32 dimensions there are more than a billion of them. |
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There is one in dimension 8 (the ''E<sub>8</sub>'' lattice), two in dimension |
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16 (''E<sub>8</sub><sup>2</sup>'' and ''II<sub>16,0</sub>''), |
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and 24 in dimension 24, called the [[Niemeier lattice]]s (examples: |
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the [[Leech lattice]], ''II<sub>24,0</sub>'', ''II<sub>16,0</sub>+II<sub>8,0</sub>'', ''II<sub>8,0</sub><sup>3</sup>''). Beyond 24 dimensions the number increases very rapidly; |
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in 32 dimensions there are more than a billion of them. |
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Unimodular lattices with no ''roots'' (vectors of norm 1 or 2) have been classified up to dimension 28. |
Unimodular lattices with no ''roots'' (vectors of norm 1 or 2) have been classified up to dimension 28. There are none of dimension less than 23 (other than the zero lattice!). |
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There are none of dimension less than 23 (other than the zero lattice!). |
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There is one in dimension 23 (called the '''short Leech lattice'''), two in dimension |
There is one in dimension 23 (called the '''short Leech lattice'''), two in dimension |
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24 (the Leech lattice and the '''odd Leech lattice'''), and {{harvtxt|Bacher|Venkov|2001}} showed that there are 0, 1, 3, 38 in dimensions |
24 (the Leech lattice and the '''odd Leech lattice'''), and {{harvtxt|Bacher|Venkov|2001}} showed that there are 0, 1, 3, 38 in dimensions 25, 26, 27, 28, respectively. Beyond this the number increases very rapidly; there are at least 8000 in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots. |
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25, 26, 27, 28. Beyond this the number increases very rapidly; there are at least 8000 |
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in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots. |
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The only non-zero example of even positive definite unimodular lattices with no |
The only non-zero example of even positive definite unimodular lattices with no roots in dimension less than 32 is the Leech lattice in dimension 24. In dimension 32 there are more than ten million examples, and above dimension 32 the number increases very rapidly. |
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roots in dimension less than 32 is the Leech lattice in dimension 24. |
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In dimension 32 there are more than ten million examples, and above dimension 32 the number increases very rapidly. |
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The following table from {{harv|King|2003}} gives the numbers of (or lower bounds for) even or odd unimodular lattices |
The following table from {{harv|King|2003}} gives the numbers of (or lower bounds for) even or odd unimodular lattices in various dimensions, and shows the very rapid growth starting shortly after dimension 24. |
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in various dimensions, and shows the very rapid growth starting shortly after dimension 24. |
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{| class="wikitable" style="margin:auto; text-align:center;" |
{| class="wikitable" style="margin:auto; text-align:center;" |
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| 15 || 5 || 0 || || |
| 15 || 5 || 0 || || |
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| 16 || 6 || 0 || 2 (E<sub>8</sub><sup>2</sup>, D<sub>16</sub><sup>+</sup>) || 0 |
| 16 || 6 || 0 || 2 (''E''<sub>8</sub><sup>2</sup>, ''D''<sub>16</sub><sup>+</sup>) || 0 |
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| 17 || 9 || 0 || || |
| 17 || 9 || 0 || || |
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| 22 || 68 || 0 || || |
| 22 || 68 || 0 || || |
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| 23 || 117 || 1 ( |
| 23 || 117 || 1 (shorter Leech lattice) || || |
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| 24 || 273 || 1 ( |
| 24 || 273 || 1 (odd Leech lattice) || 24 (Niemeier lattices) || 1 (Leech lattice) |
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| 25 || 665 || 0 || || |
| 25 || 665 || 0 || || |
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| 26 || |
| 26 || ≥ 2307 || 1 || || |
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| 27 || |
| 27 || ≥ 14179 || 3 || || |
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| 28 || |
| 28 || ≥ 327972 || 38 || || |
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| 29 || |
| 29 || ≥ 37938009 || ≥ 8900 || || |
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| 30 || |
| 30 || ≥ 20169641025 || ≥ 82000000 || || |
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| 31 || |
| 31 || ≥ 5x10<sup>12</sup> || ≥ 8×10<sup>11</sup> || || |
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| 32 || |
| 32 || ≥ 8x10<sup>16</sup> || ≥ 1×10<sup>16</sup> || ≥ 1162109024 || ≥ 10000000 |
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== Applications == |
== Applications == |
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The second [[cohomology group]] of a closed [[simply connected]] [[oriented]] topological [[4-manifold]] is a unimodular lattice. [[Michael Freedman]] showed that this lattice almost determines the manifold: there is a unique such manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice. In particular if we take the lattice to be 0, this implies the [[Poincaré conjecture]] for 4-dimensional topological manifolds. [[Donaldson's theorem]] states that if the manifold is [[Smooth manifold|smooth]] and the lattice is positive definite, then it must be a sum of copies of '''Z''', so most of these manifolds have no [[smooth structure]]. |
The second [[cohomology group]] of a closed [[simply connected]] [[oriented]] topological [[4-manifold]] is a unimodular lattice. [[Michael Freedman]] showed that this lattice almost determines the [[manifold]]: there is a unique such manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice. In particular if we take the lattice to be 0, this implies the [[Poincaré conjecture]] for 4-dimensional topological manifolds. [[Donaldson's theorem]] states that if the manifold is [[Smooth manifold|smooth]] and the lattice is positive definite, then it must be a sum of copies of '''Z''', so most of these manifolds have no [[smooth structure]]. One such example is the [[E8 manifold|<math>E_8</math> manifold]]. |
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== References == |
== References == |
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{{Reflist}} |
{{Reflist}} |
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⚫ | *{{citation | last1=Bacher | first1=Roland | last2=Venkov | first2=Boris | chapter-url=http://www-fourier.ujf-grenoble.fr/PREP/html/a332/a332.html | chapter=Réseaux entiers unimodulaires sans racine en dimension 27 et 28 | trans-chapter=Unimodular integral lattices without roots in dimensions 27 and 28 | title=Réseaux euclidiens, designs sphériques et formes modulaires | trans-title=Euclidean lattices, spherical designs and modular forms | language=fr | editor-last=Martinet | editor-first=Jacques | pages=212–267 | series=Monogr. Enseign. Math. | volume=37 | location=Geneva | year=2001 | isbn=2-940264-02-3 | mr=1878751 | zbl=1139.11319 | publisher=L'Enseignement Mathématique | url-status=dead | archive-url=https://web.archive.org/web/20070928015916/http://www-fourier.ujf-grenoble.fr/PREP/html/a332/a332.html | archive-date=2007-09-28 }} |
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*{{citation|last=Bacher|first= Roland|last2= Venkov|first2= Boris |chapter-url=http://www-fourier.ujf-grenoble.fr/PREP/html/a332/a332.html |
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⚫ | *{{citation | zbl=0915.52003 | last1=Conway | first1=J.H. | author1-link=John Horton Conway | last2=Sloane | first2=N.J.A. | author2-link=Neil Sloane | others=With contributions by Bannai, E.; Borcherds, R.E.; Leech, J.; Norton, S.P.; Odlyzko, A.M.; Parker, R.A.; Queen, L.; Venkov, B.B. | title=Sphere packings, lattices and groups | edition=Third | series=Grundlehren der Mathematischen Wissenschaften | volume=290 | location=New York, NY | publisher=[[Springer-Verlag]] | isbn=0-387-98585-9 | mr=662447 | year=1999 | url-access=registration | url=https://archive.org/details/spherepackingsla0000conw_b8u0 }} |
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|chapter=Réseaux entiers unimodulaires sans racine en dimension 27 et 28 | trans-chapter=Unimodular integral lattices without roots in dimensions 27 and 28 |
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⚫ | *{{Citation | last1=King | first1=Oliver D. | title=A mass formula for unimodular lattices with no roots | arxiv=math.NT/0012231 | doi=10.1090/S0025-5718-02-01455-2 | mr=1954971 | year=2003 | journal=[[Mathematics of Computation]] | volume=72 | issue=242 | pages=839–863 | zbl=1099.11035 | bibcode=2003MaCom..72..839K | s2cid=7766244 }} |
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⚫ | |||
⚫ | *{{citation | zbl=0915.52003 | last1=Conway | first1=J.H. | author1-link=John Horton Conway | last2=Sloane | first2=N.J.A. | author2-link=Neil Sloane | others=With contributions by Bannai, E.; Borcherds, R.E.; Leech, J.; Norton, S.P.; Odlyzko, A.M.; Parker, R.A.; Queen, L.; Venkov, B.B. | title=Sphere packings, lattices and groups | edition=Third | series=Grundlehren der Mathematischen Wissenschaften | volume=290 | location=New York, NY | publisher=[[Springer-Verlag]] | isbn=0-387-98585-9 | mr=662447 | year=1999}} |
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⚫ | |||
* {{citation | first1=John | last1=Milnor | author1-link=John Milnor| first2=Dale | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 | mr=0506372 | location=New York-Heidelberg | doi=10.1007/978-3-642-88330-9}} |
* {{citation | first1=John | last1=Milnor | author1-link=John Milnor| first2=Dale | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 | mr=0506372 | location=New York-Heidelberg | doi=10.1007/978-3-642-88330-9}} |
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* {{citation | first=Jean-Pierre | last=Serre | |
* {{citation | first=Jean-Pierre | last=Serre | author-link=Jean-Pierre Serre | title=A Course in Arithmetic | series=[[Graduate Texts in Mathematics]] | volume=7 | publisher=[[Springer-Verlag]] | year=1973 | isbn=0-387-90040-3 | zbl=0256.12001 | doi=10.1007/978-1-4684-9884-4 | mr=0344216 | url-access=registration | url=https://archive.org/details/courseinarithmet00serr }} |
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* {{citation| first = Michael H. | last = Freedman | author-link=Michael Freedman | title= The topology of four-dimensional manifolds | journal = J. Differential Geom. | number =3 | volume = 17 | pages = 357–453| year = 1982 | doi = 10.4310/jdg/1214437136 | doi-access = free }} |
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==External links== |
==External links== |
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*[[Neil Sloane]]'s [http://www. |
*[[Gabriele Nebe]] and [[Neil Sloane]]'s [http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/unimodular.html catalogue] of unimodular lattices. |
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*{{ |
*{{OEIS el|sequencenumber=A005134|name=Number of n-dimensional unimodular lattices|formalname=Number of n-dimensional unimodular lattice (or quadratic forms)}} |
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[[Category:Quadratic forms]] |
[[Category:Quadratic forms]] |
Latest revision as of 08:03, 16 July 2024
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1.
The E8 lattice and the Leech lattice are two famous examples.
Definitions
[edit]- A lattice is a free abelian group of finite rank with a symmetric bilinear form (·, ·).
- The lattice is integral if (·,·) takes integer values.
- The dimension of a lattice is the same as its rank (as a Z-module).
- The norm of a lattice element a is (a, a).
- A lattice is positive definite if the norm of all nonzero elements is positive.
- The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (ai, aj), where the elements ai form a basis for the lattice.
- An integral lattice is unimodular if its determinant is 1 or −1.
- A unimodular lattice is even or type II if all norms are even, otherwise odd or type I.
- The minimum of a positive definite lattice is the lowest nonzero norm.
- Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is.
- The signature of a lattice is the signature of the form on the vector space.
Examples
[edit]The three most important examples of unimodular lattices are:
- The lattice Z, in one dimension.
- The E8 lattice, an even 8-dimensional lattice,
- The Leech lattice, the 24-dimensional even unimodular lattice with no roots.
Properties
[edit]An integral lattice is unimodular if and only if its dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual.
Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m−n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8. Examples in all admissible signatures are given by the IIm,n and Im,n constructions, respectively.
The theta function of a unimodular positive definite lattice is a modular form whose weight is one half the rank. If the lattice is even, the form has level 1, and if the lattice is odd the form has Γ0(4) structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣n/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80,[1] and their non-existence has been proven for dimensions above 163,264.[2]
Classification
[edit]For indefinite lattices, the classification is easy to describe. Write Rm,n for the m + n dimensional vector space Rm+n with the inner product of (a1, ..., am+n) and (b1, ..., bm+n) given by
In Rm,n there is one odd indefinite unimodular lattice up to isomorphism, denoted by
- Im,n,
which is given by all vectors (a1,...,am+n) in Rm,n with all the ai integers.
There are no indefinite even unimodular lattices unless
- m − n is divisible by 8,
in which case there is a unique example up to isomorphism, denoted by
- IIm,n.
This is given by all vectors (a1,...,am+n) in Rm,n such that either all the ai are integers or they are all integers plus 1/2, and their sum is even. The lattice II8,0 is the same as the E8 lattice.
Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example In,0 in each dimension n less than 8, and two examples (I8,0 and II8,0) in dimension 8. The number of lattices increases moderately up to dimension 25 (where there are 665 of them), but beyond dimension 25 the Smith-Minkowski-Siegel mass formula implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 in dimension 32.
In some sense unimodular lattices up to dimension 9 are controlled by E8, and up to dimension 25 they are controlled by the Leech lattice, and this accounts for their unusually good behavior in these dimensions. For example, the Dynkin diagram of the norm-2 vectors of unimodular lattices in dimension up to 25 can be naturally identified with a configuration of vectors in the Leech lattice. The wild increase in numbers beyond 25 dimensions might be attributed to the fact that these lattices are no longer controlled by the Leech lattice.
Even positive definite unimodular lattice exist only in dimensions divisible by 8. There is one in dimension 8 (the E8 lattice), two in dimension 16 (E82 and II16,0), and 24 in dimension 24, called the Niemeier lattices (examples: the Leech lattice, II24,0, II16,0 + II8,0, II8,03). Beyond 24 dimensions the number increases very rapidly; in 32 dimensions there are more than a billion of them.
Unimodular lattices with no roots (vectors of norm 1 or 2) have been classified up to dimension 28. There are none of dimension less than 23 (other than the zero lattice!). There is one in dimension 23 (called the short Leech lattice), two in dimension 24 (the Leech lattice and the odd Leech lattice), and Bacher & Venkov (2001) showed that there are 0, 1, 3, 38 in dimensions 25, 26, 27, 28, respectively. Beyond this the number increases very rapidly; there are at least 8000 in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots.
The only non-zero example of even positive definite unimodular lattices with no roots in dimension less than 32 is the Leech lattice in dimension 24. In dimension 32 there are more than ten million examples, and above dimension 32 the number increases very rapidly.
The following table from (King 2003) gives the numbers of (or lower bounds for) even or odd unimodular lattices in various dimensions, and shows the very rapid growth starting shortly after dimension 24.
Dimension | Odd lattices | Odd lattices no roots |
Even lattices | Even lattices no roots |
---|---|---|---|---|
0 | 0 | 0 | 1 | 1 |
1 | 1 | 0 | ||
2 | 1 | 0 | ||
3 | 1 | 0 | ||
4 | 1 | 0 | ||
5 | 1 | 0 | ||
6 | 1 | 0 | ||
7 | 1 | 0 | ||
8 | 1 | 0 | 1 (E8 lattice) | 0 |
9 | 2 | 0 | ||
10 | 2 | 0 | ||
11 | 2 | 0 | ||
12 | 3 | 0 | ||
13 | 3 | 0 | ||
14 | 4 | 0 | ||
15 | 5 | 0 | ||
16 | 6 | 0 | 2 (E82, D16+) | 0 |
17 | 9 | 0 | ||
18 | 13 | 0 | ||
19 | 16 | 0 | ||
20 | 28 | 0 | ||
21 | 40 | 0 | ||
22 | 68 | 0 | ||
23 | 117 | 1 (shorter Leech lattice) | ||
24 | 273 | 1 (odd Leech lattice) | 24 (Niemeier lattices) | 1 (Leech lattice) |
25 | 665 | 0 | ||
26 | ≥ 2307 | 1 | ||
27 | ≥ 14179 | 3 | ||
28 | ≥ 327972 | 38 | ||
29 | ≥ 37938009 | ≥ 8900 | ||
30 | ≥ 20169641025 | ≥ 82000000 | ||
31 | ≥ 5x1012 | ≥ 8×1011 | ||
32 | ≥ 8x1016 | ≥ 1×1016 | ≥ 1162109024 | ≥ 10000000 |
Beyond 32 dimensions, the numbers increase even more rapidly.
Applications
[edit]The second cohomology group of a closed simply connected oriented topological 4-manifold is a unimodular lattice. Michael Freedman showed that this lattice almost determines the manifold: there is a unique such manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice. In particular if we take the lattice to be 0, this implies the Poincaré conjecture for 4-dimensional topological manifolds. Donaldson's theorem states that if the manifold is smooth and the lattice is positive definite, then it must be a sum of copies of Z, so most of these manifolds have no smooth structure. One such example is the manifold.
References
[edit]- ^ Nebe, Gabriele; Sloane, Neil. "Unimodular Lattices, Together With A Table of the Best Such Lattices". Online Catalogue of Lattices. Retrieved 2015-05-30.
- ^ Nebe, Gabriele (2013). "Boris Venkov's Theory of Lattices and Spherical Designs". In Wan, Wai Kiu; Fukshansky, Lenny; Schulze-Pillot, Rainer; et al. (eds.). Diophantine methods, lattices, and arithmetic theory of quadratic forms. Contemporary Mathematics. Vol. 587. Providence, RI: American Mathematical Society. pp. 1–19. arXiv:1201.1834. Bibcode:2012arXiv1201.1834N. MR 3074799.
- Bacher, Roland; Venkov, Boris (2001), "Réseaux entiers unimodulaires sans racine en dimension 27 et 28" [Unimodular integral lattices without roots in dimensions 27 and 28], in Martinet, Jacques (ed.), Réseaux euclidiens, designs sphériques et formes modulaires [Euclidean lattices, spherical designs and modular forms], Monogr. Enseign. Math. (in French), vol. 37, Geneva: L'Enseignement Mathématique, pp. 212–267, ISBN 2-940264-02-3, MR 1878751, Zbl 1139.11319, archived from the original on 2007-09-28
- Conway, J.H.; Sloane, N.J.A. (1999), Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, vol. 290, With contributions by Bannai, E.; Borcherds, R.E.; Leech, J.; Norton, S.P.; Odlyzko, A.M.; Parker, R.A.; Queen, L.; Venkov, B.B. (Third ed.), New York, NY: Springer-Verlag, ISBN 0-387-98585-9, MR 0662447, Zbl 0915.52003
- King, Oliver D. (2003), "A mass formula for unimodular lattices with no roots", Mathematics of Computation, 72 (242): 839–863, arXiv:math.NT/0012231, Bibcode:2003MaCom..72..839K, doi:10.1090/S0025-5718-02-01455-2, MR 1954971, S2CID 7766244, Zbl 1099.11035
- Milnor, John; Husemoller, Dale (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, New York-Heidelberg: Springer-Verlag, doi:10.1007/978-3-642-88330-9, ISBN 3-540-06009-X, MR 0506372, Zbl 0292.10016
- Serre, Jean-Pierre (1973), A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7, Springer-Verlag, doi:10.1007/978-1-4684-9884-4, ISBN 0-387-90040-3, MR 0344216, Zbl 0256.12001
- Freedman, Michael H. (1982), "The topology of four-dimensional manifolds", J. Differential Geom., 17 (3): 357–453, doi:10.4310/jdg/1214437136
External links
[edit]- Gabriele Nebe and Neil Sloane's catalogue of unimodular lattices.
- OEIS sequence A005134 (Number of n-dimensional unimodular lattices)