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{| class=wikitable width=280 align=right
In mathematics, the '''Kervaire semi-characteristic''', introduced by {{harvs|txt|last=Kervaire|authorlink=Michel Kervaire|year=1956}}, is an invariant of manifolds ''M''  of dimension 4''n''+1 taking values in '''Z'''/2'''Z''', given by
!<math>{\tilde{A}}_2</math>
:k(M) = <math>\sum_{i=0}^n \dim H^{2i}(M,R)\bmod 2</math>
!<math>{\tilde{A}}_3</math>
{{harvtxt|Atiyah|Singer|1971}} showed that it is given by the index of a skew-adjoint elliptic operator.
|-
![[Triangular tiling]]
![[Tetrahedral-octahedral honeycomb]]
|-
|[[File:Uniform_tiling_333-t1.png|120px]]<BR>With red and yellow equilateral triangles
|[[File:Tetrahedral-octahedral honeycomb2.png|160px]]<BR>With cyan and yellow [[tetrahedron|tetrahedra]], and red rectified tetrahedra ([[octahedron]])
|-
!{{CDD|node_1|split1|branch}}
!{{CDD|node_1|split1|nodes|split2|node}}
|}
In [[geometry]], the '''simplectic honeycomb''' (or '''n-simplex honeycomb''') is a dimensional infinite series of [[Honeycomb (geometry)|honeycomb]]s, based on the <math>{\tilde{A}}_n</math> affine [[Coxeter group]] symmetry. It is given a [[Schläfli symbol]] {3<sup>[n+1]</sup>}, and is represented by a [[Coxeter-Dynkin diagram]] as a cyclic graph of ''n+1'' nodes with one node ringed. It is composed of n-[[simplex]] facets, along with all [[Rectification (geometry)|rectified]] n-simplices. The [[vertex figure]] of an ''n-simplex honeycomb'' is an [[Expansion (geometry)|expanded]] n-[[simplex]].


In 2 dimensions, the honeycomb represents the [[triangular tiling]], with Coxeter graph {{CDD|node_1|split1|branch}} filling the plane with alternately colored triangles. In 3 dimensions it represents the [[tetrahedral-octahedral honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|split2|node}} filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the [[5-cell honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|branch}}, with [[5-cell]] and [[rectified 5-cell]] facets. In 5 dimensions it is called the [[5-simplex honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}, filling space by [[5-simplex]], [[rectified 5-simplex]], and [[birectified 5-simplex]] facets. In 6 dimensions it is called the [[6-simplex honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}, filling space by [[6-simplex]], [[rectified 6-simplex]], and [[birectified 6-simplex]] facets.
==References==


== By dimension ==
*{{citation|last1= Atiyah|first1= Michael F. |author1-link=Michael Atiyah|last2=Singer|first2= Isadore M. |author2-link=Isadore Singer|title=The Index of Elliptic Operators V|journal=[[Annals of Mathematics. Second Series]]|volume= 93|issue= 1|year= 1971|pages= 139–149|doi= 10.2307/1970757|publisher= The Annals of Mathematics, Vol. 93, No. 1 |jstor=1970757}}
{| class="wikitable"
*{{Citation | last1=Kervaire | first1=Michel | title=Courbure intégrale généralisée et homotopie | doi=10.1007/BF01342961 |mr=0086302 | year=1956 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=131 | pages=219–252}}
!height=30|n
!<math>{\tilde{A}}_{2+}</math>
!Tessellation
!Vertex figure
!Facets per vertex figure
!Vertices per vertex figure
!Edge figure
|-
|1
|<math>{\tilde{A}}_1</math>
|[[File:Regular_apeirogon.png|80px]]<BR>[[Apeirogon]]<BR>{{CDD|node_1|infin|node}}
|{{CDD|node_1}}
|1
|2
| -
|-
|2
|<math>{\tilde{A}}_2</math>
|[[Image:Uniform tiling 333-t1.png|80px]]<BR>[[Triangular tiling]]<BR>2-simplex honeycomb<BR>{{CDD|node_1|split1|branch}}
|[[Image:Truncated triangle.png|80px]]<BR>[[Hexagon]]<BR>(Truncated triangle)<BR>{{CDD|node_1|3|node_1}}
|3 [[triangle]]s<BR>3 [[hexagon|rectified triangles]]
|6
|[[Line segment]]<BR>{{CDD|node_1}}
|-
|3
|<math>{\tilde{A}}_3</math>
|[[File:Tetrahedral-octahedral honeycomb2.png|80px]]<BR>[[Tetrahedral-octahedral honeycomb]]<BR>3-simplex honeycomb<BR>{{CDD|node_1|split1|nodes|split2|node}}
|[[Image:Uniform polyhedron-33-t02.png|80px]]<BR>[[Cuboctahedron]]<BR>(Cantellated tetrahedron)<BR>{{CDD|node_1|3|node|3|node_1}}
|4+4 [[tetrahedron]]<BR>6 [[octahedron|rectified tetrahedra]]
|12
|[[Rectangle]]<BR>{{CDD|node_1|2|node_1}}
|-
|4
|<math>{\tilde{A}}_4</math>
|[[4-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|branch}}
|[[Image:Schlegel half-solid runcinated 5-cell.png|80px]]<BR>[[Runcinated 5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node_1}}
|5+5 [[5-cell]]s<BR>10+10 [[rectified 5-cell]]s
|20
|[[File:Runcinated_5-cell_verf.png|60px]]<BR>Triangular antiprism<BR>{{CDD|node_h|3|node_h|2|node_h}}
|-
|5
|<math>{\tilde{A}}_5</math>
|[[5-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
|[[File:5-simplex_t04.svg|80px]]<BR>[[Stericated 5-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1}}
|6+6 [[5-simplex]]<BR>15+15 [[rectified 5-simplex]]<BR>20 [[birectified 5-simplex]]
|30
|[[File:Stericated_hexateron_verf.png|60px]]<BR>Tetrahedral antiprism<BR>{{CDD|node_h|4|node|3|node|2|node_h}}
|-
|6
|<math>{\tilde{A}}_6</math>
|[[6-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}
|[[File:6-simplex_t05.svg|80px]]<BR>[[Pentellated 6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1}}
|7+7 [[6-simplex]]<BR>21+21 [[rectified 6-simplex]]<BR>35+35 [[birectified 6-simplex]]
|42
|4-simplex antiprism
|-
|7
|<math>{\tilde{A}}_7</math>
|[[7-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
|[[File:7-simplex_t06.svg|80px]]<BR>[[Hexicated 7-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|8+8 [[7-simplex]]<BR>28+28 [[rectified 7-simplex]]<BR>56+56 [[birectified 7-simplex]]<BR>70 [[trirectified 7-simplex]]
|56
|5-simplex antiprism
|-
|8
|<math>{\tilde{A}}_8</math>
|[[8-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
|[[File:8-simplex_t07.svg|80px]]<BR>[[Heptellated 8-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|9+9 [[8-simplex]]<BR>36+36 [[rectified 8-simplex]]<BR>84+84 [[birectified 8-simplex]]<BR>126+126 [[trirectified 8-simplex]]
|72
|6-simplex antiprism
|-
|9
|<math>{\tilde{A}}_9</math>
|[[9-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
|[[File:9-simplex_t08.svg|80px]]<BR>[[Octellated 9-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|10+10 [[9-simplex]]<BR>45+45 [[rectified 9-simplex]]<BR>120+120 [[birectified 9-simplex]]<br>210+210 [[trirectified 9-simplex]]<br>252 [[quadrirectified 9-simplex]]
|90
|7-simplex antiprism
|-
|10
|<math>{\tilde{A}}_{10}</math>
|[[10-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
|[[File:10-simplex_t09.svg|80px]]<BR>[[Ennecated 10-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|11+11 [[10-simplex]]<BR>55+55 [[rectified 10-simplex]]<BR>165+165 [[birectified 10-simplex]]<BR>330+330 [[trirectified 10-simplex]]<BR>462+462 [[quadrirectified 10-simplex]]
|110
|8-simplex antiprism
|-
|11
|<math>{\tilde{A}}_{11}</math>
|11-simplex honeycomb
|...
|...
|...
|...
|}


== Projection by folding ==
[[Category:Differential topology]]
 
The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional [[hypercubic honeycomb]] by a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] operation that maps two pairs of mirrors into each other, sharing the same [[vertex arrangement]]:
 
{|class=wikitable
|-
!<math>{\tilde{A}}_2</math>
|{{CDD|node_1|split1|branch}}
!<math>{\tilde{A}}_4</math>
|{{CDD|node_1|split1|nodes|3ab|branch}}
!<math>{\tilde{A}}_6</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}
!<math>{\tilde{A}}_8</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
!<math>{\tilde{A}}_{10}</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
|...
 
|-
!<math>{\tilde{A}}_3</math>
|{{CDD|nodes_10r|splitcross|nodes}}
!<math>{\tilde{A}}_3</math>
|{{CDD|node_1|split1|nodes|split2|node}}
!<math>{\tilde{A}}_5</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
!<math>{\tilde{A}}_7</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
!<math>{\tilde{A}}_9</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
|...
|-
!<math>{\tilde{C}}_1</math>
|{{CDD|node_1|infin|node}}
!<math>{\tilde{C}}_2</math>
|{{CDD|node_1|4|node|4|node}}
!<math>{\tilde{C}}_3</math>
|{{CDD|node_1|4|node|3|node|4|node}}
!<math>{\tilde{C}}_4</math>
|{{CDD|node_1|4|node|3|node|3|node|4|node}}
!<math>{\tilde{C}}_5</math>
|{{CDD|node_1|4|node|3|node|3|node|3|node|4|node}}
|...
|}
 
== Kissing number ==
 
These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the [[vertex figure]]. For 2 and 3 dimensions, this represents the highest [[kissing number]] for 2 and 3 dimensions, but fall short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in an [[cuboctahedron|cuboctahedral]] configuration. For 4 to 8 dimensions, the kissing numbers are [[Expanded 4-simplex|20]], [[Expanded 5-simplex|30]], [[Expanded 5-simplex|42]], [[Expanded 6-simplex|56]], and [[Expanded 7-simplex|72]] spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.
 
== See also==
* [[Truncated simplectic honeycomb]]
* [[Omnitruncated simplectic honeycomb]]
 
== References ==
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
* [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
 
{{Honeycombs}}
 
[[Category:Honeycombs (geometry)]]
[[Category:Polytopes]]

Revision as of 19:08, 18 August 2014

In mathematics, the Kervaire semi-characteristic, introduced by Template:Harvs, is an invariant of manifolds M of dimension 4n+1 taking values in Z/2Z, given by

k(M) = i=0ndimH2i(M,R)mod2

Template:Harvtxt showed that it is given by the index of a skew-adjoint elliptic operator.

References

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