Eight in algebra, topology and mathematical physics

A collection of resources on the occurrences of the number 8 in algebra, topology and mathematical physics, related to quadratic forms (especially E8) and the topology of manifolds.
The papers are arranged chronologically.
Please email me at a.ranicki@ed.ac.uk with suggestions for further inclusions.

  • The number 8 in Sesame Street

  • Wikipedia article on the E_8 lattice

  • H.J.S. Smith, On the Orders and Genera of Ternary Quadratic Forms. Proc. Royal Society of London 16, 197-208 (1867)
    Primordial appearence of the rank 8 symmetric positive definite integral form E8
  • A. Korkine and G. Zolotareff, Sur les formes quadratiques positives. Mathematische Annalen 6, 242-292 (1873)
  • T. Gosset, On the regular and semi-regular figures in space of n dimensions, Messenger of Mathematics 29, 43-48 (1900)
  • Wikipedia articles on Thorold Gosset and his 4_21 polytope
  • W. Boy, Über die Curvatura integra und die Topologie geschlossener Flächen Math. Ann. 57, 151-184 (1903)
    The original immersion of RP2 in R3. Boy's immersion generates the cobordism group of immersions of surfaces in R3
    which is isomorphic to Z8. More about Boy's immersion here.
  • L.J. Mordell, The definite quadratic forms in eight variables with determinant unity. Journ de math. XVII, 41--46 (1938)
  • H. Braun, Geschlechter quadratischer Formen, J. Reine und Angew. Math. 182, 32-49 (1940)
  • J. Milnor, On simply-connected 4-manifolds Proc. 1956 Algebraic Topology Conference, Mexico 122-128, (1958)
  • F. van der Blij, An invariant of quadratic forms mod 8, Indag. Math. 21, 291-293 (1959)
  • W. Ledermann, An arithmetical property of quadratic forms, Comment. Math. Helv. 33, 34--37 (1959)
  • F. Hirzebruch, Zur Theorie der Mannigfaltigkeiten, Int. Math. Nachrichten 67, 50 (1961)
  • J.-P. Serre, Formes bilineaires symetriques entieres a discriminant +/- 1 Seminaire Cartan, 1961/62, Exp. 14-15 (1961/1962)
  • M.Kervaire and J. Milnor, Groups of homotopy spheres I. Annals of Maths. 77, 504--537 (1963)
  • A. Liulevicius, A theorem in homological algebra and stable homotopy of projective spaces. Trans. Amer. Math. Soc. 109, 540-552 (1963)
  • C.T.C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213, 187-199 (1964)
  • M. Atiyah, R. Bott and A. Shapiro Clifford modules Topology 3, 3-38 (1964)
  • R. Wells, Cobordism groups of immersions Topology 5, 281-294 (1966)
  • E.H. Brown, The Kervaire invariant of a manifold Proc. Sympos. Pure Math., Vol. XXII, AMS, 65-71 (1971)
  • L. Siebenmann, Topological manifolds, Proc. 1970 Nice ICM, Vol.2, 133-163 (1971)
  • S. Morita, On the Pontrjagin square and the signature, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18, 405-414 (1971)
  • E.H. Brown, Generalizations of the Kervaire invariant Ann. of Maths. 95, 368-383 (1972)
  • G. Brumfiel and J. Morgan, Quadratic functions, the index modulo 8, and a Z/4-Hirzebruch formula, Topology 12, 105-122 (1973)
  • D. Frank, The signature mod 8, Comment. Math. Helv. 48, 520-524 (1973)
  • J. Milnor and D. Husemoller, Symmetric Bilinear Forms (extracts), Springer (1973)
  • J. Barge, J. Lannes, F. Latour and P. Vogel, Λ-sphères Ann. Sci. École Norm. Sup. (4) 7, 463-505 (1974)
  • J. Morgan and D. Sullivan, The transversality characteristic class and linking cycles in surgery theory, Ann. of Math. (2) 99, 463-544 (1974)
  • J. Milgram, Surgery with coefficients, Ann. of Math. (2) 100, 194-248 (1974)
  • J. Lannes and F. Latour, Forme quadratique d'enlacement et applications, Asterisque No. 26, SMF (1975)
  • J. Lannes, Formes quadratiques d'enlacement sur l'anneau des entiers d'un corps de nombres, Ann. Sci. École Norm. Sup. (4) 8, 535-579 (1975)
  • G. Brumfiel and J. Morgan, Homotopy theoretic consequences of N.Levitt's obstruction theory to transversality for spherical fibrations, Pacific J. Math.67, 1-100 (1976)
  • A. Phillips, Eight Boys bound, Supplementary material Videos (1977)
  • R. Kirby and M. Scharlemann, Eight faces of the Poincare homology 3-sphere Geometric topology (Georgia Topology Conf. 1977), pp. 113-146 (1979)
  • A. Ranicki, The total surgery obstruction, Springer Lecture Notes 763, 275--316 (1979)
  • N. Habegger, Une variete de dimension 4 avec forme d'intersection paire et signature -8, Comment. Math. Helv. 57, 22-24 (1982)
  • R. Cohen, J. Jones and M. Mahowald, The Kervaire invariant of immersions, Invent. Math. 79, 95-123 (1985)
  • E. Witten, Global gravitational anomalies, Comm. Math. Phys. 100, 197-229 (1985)
  • U. Pinkall, Regular homotopy classes of immersed surfaces Topology 24, 421-434 (1985)
  • M. Weiss, Surgery and the generalized Kervaire invariant, Proc. London Math. Soc. (3) 51, I. 146--192, II. 193--230 (1985)
  • W. Scharlau, Hermite's Inequality and the Form E8, Appendix 4 of "Quadratic and Hermitian Forms", Springer (1985)
  • L. Guillou and A. Marin, A la recherche du topologie perdue (Rohlin), Birkhauser (1986)
  • T. Matumoto, An elementary proof of Rochlin's signature theorem and its extension by Guillou and Marin A la recherche du topologie perdue (Rohlin), pp. 119-139, Birkhauser (1986)
  • N. Secord, Demodulation decoding of a DC free construction of the Gosset lattice, McMaster Ph.D. thesis (1989)
  • R. Kirby and L. Taylor, Pin structures on low-dimensional manifolds LMS Lecture Notes 151, 177-242 (1990)
  • M. Freedman and F. Quinn, Classifications and embeddings, Chapter 10 of Topology of 4-manifolds, Princeton (1990)
  • M. de Gosson, Maslov indices on the metaplectic group Mp(n), Ann. Inst. Fourier. Grenoble 40, 537-555 (1990)
  • S. Ochanine, Elliptic genera, modular forms over KO and the Brown-Kervaire invariant., Math. Z. 206, 277-291 (1991)
  • A. Ranicki, Algebraic L-theory and topological manifolds, Cambridge (1992)
  • M. Weiss, Visible L-theory, Forum Math. 4, 465-498 (1992)
  • J. Barge and E. Ghys, Cocycles d'Euler et de Maslov, Math. Ann. 294, 235--265 (1992)
  • J. Wood,Witt's extension theorem for mod four valued quadratic forms. Trans. Amer. Math. Soc. 336, 445-461 (1993)
  • R. Kirby and P. Melvin, Dedekind sums, μ-invariants and the signature cocycle, Math. Ann. 299, 231-267 (1994)
  • S. Klaus, Brown-Kervaire invariants, Mainz Ph.D. thesis (1995)
  • P. Deligne, Notes on spinors (1996)
  • J. Conway and N.J.A. Sloane, On the classification of integral quadratic forms Chapter 15 of Sphere Packings, Lattices and Groups, Springer (1998)
  • I. Naruki and N. Tsuchida, On the signature for finite quadratic forms Mem. Inst. Sci. Engrg. Ritsumeikan Univ. 57, 33-48 (1998)
  • R. Griess, Pieces of eight: semiselfdual lattices and a new foundation for the theory of Conway and Mathieu groups, Adv. Math. 148, 75-104 (1999).
  • L. Taylor, Gauss Sums in Algebra and Topology (2000?)
  • A. Degtyarev, On the Pontryagin-Viro form, in Topology, ergodic theory, real algebraic geometry, AMS Transl. Ser. 2, 202, 71--94 (2001)
  • A. Ranicki, Algebraic Poincare cobordism, Contemporary Mathematics 279, A.M.S., 213--255 (2001)
  • L. Funar and R. Gini, The graded cobordism group of codimension-one immersions, Geometric and Functional Analysis 12, 1235--1264 (2002)
  • J. Baez, The octonions, Bull. A.M.S. 39, 145-205 (2002)
  • A. Szucs, Two theorems of Rokhlin J. Math. Sci. (N. Y.) 113, 888-892 (2003)
  • R. Kirby and P. Melvin, Local surgery formulas for quantum invariants and the Arf invariant Proc. CassonFest, Geom. Topol. Monogr. 7, 213-233 (2004)
  • D. Richter, Gosset's figure in a Clifford algebra Adv. Appl. Clifford Algebr. 14, 215--224 (2004)
  • A. Ranicki, The signature mod 2, 4 and 8, Slides of talk in Oxford on joint work with Larry Taylor (2005)
  • F. Deloup and G. Massuyeau, Quadratic functions on torsion groups J. Pure Appl. Algebra 198, 105-121 (2005)
  • M. Hopkins and I.Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70, 329-452 (2005)
  • A. Scorpan, The wild world of 4-manifolds (Chapters 3,4 only), AMS (2005)
  • W. Mio and A. Ranicki, The quadratic form E_8 and exotic homology manifolds, Geometry and Topology Monograph 9, 33--66 (2006)
  • M. Banagl and A. Ranicki, Generalized Arf invariants in algebraic L-theory Advances in Mathematics 199, 542-668 (2006)
  • A. Ranicki and L. Taylor, The signature mod 2, 4 and 8, Slides for topology seminar in Oxford (2006)
  • K-U. Schmidt, Z_4-valued quadratic forms and quaternary sequence families, IEEE Trans. Inform. Theory 55, 5803-5810 (2009)
  • N. Li, X. Tang and T. Helleseth, Several classes of codes and sequences derived from a Z_4-valued quadratic form, IEEE Trans. Inform. Theory 57, 7618-7628 (2011)
  • M. Hill, M. Hopkins and D. Ravenel, On the non-existence of elements of Kervaire invariant one Arxiv 0908.3724 (2009-2014)
  • H. Sati, Topological aspects of the NS 5-brane, Arxiv 1109.4834 (2011-2014)
  • S. Gurevich and R. Hadani The Weil representation in characteristic two, Advances in Mathematics 230, 894--926 (2012}
  • J. Li, X. Zeng, X. Tang, C. Li, A family of quadriphase sequences of period 4(2^n-1) with low correlation and large linear span, Des. Codes Cryptogr. 67, 19-35 (2013)
  • L. Funar and W. Pitsch, Finite quotients of symplectic groups vs mapping class groups, Arxiv (2013)
  • N. Li, X. Tang and T. Helleseth, New constructions of quadratic bent functions in polynomial form, IEEE Trans. Inform. Theory 60, 5760-5767 (2014)
  • R. Pendavingh, On the evaluation at (i,-i) of the Tutte polynomial of a binary matroid, J. Algebraic Combin. 39, 141-152 (2014)
  • A. Phillips, Immersed surfaces I. (Video of lecture at Stony Brook, 19.12.2014)
  • A. Ranicki, Immersed surfaces II. (Video of lecture at Stony Brook, 19.12.2014)
  • A. Ranicki, The number eight in topology, Slides of lecture at ECSTATIC conference, Imperial College London 11.6.2015
  • S. Garibaldi, E8, the most exceptional group. http://arxiv.org/abs/1605.01721
  • M. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), 991-1015.