Illinois Journal of Mathematics

The string bordism of $BE_8$ and $BE_8\times BE_8$ through dimension $14$

Michael A. Hill

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We compute the low dimensional String bordism groups $\widetilde{\Omega}_{k}^{\mathit{String}}BE_8$ and $\widetilde{\Omega }_{k}^{\mathit{String}}(BE_8\times BE_8)$ using a combination of Adams spectral sequences together with comparisons to the Spin bordism cases.

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Illinois J. Math., Volume 53, Number 1 (2009), 183-196.

First available in Project Euclid: 22 January 2010

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Primary: 57R90: Other types of cobordism [See also 55N22] 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 55N34: Elliptic cohomology


Hill, Michael A. The string bordism of $BE_8$ and $BE_8\times BE_8$ through dimension $14$. Illinois J. Math. 53 (2009), no. 1, 183--196. doi:10.1215/ijm/1264170845.

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  • D. W. Anderson, E. H. Brown, Jr. and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298.
  • M. Ando, M. J. Hopkins and C. Rezk, Multiplicative orientations of ${K}{O}$-theory and of the spectrum of topological modular forms, in preparation.
  • M. Ando, M. J. Hopkins and N. P. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001), 595–687.
  • M. Ando, M. J. Hopkins and N. P. Strickland, The sigma orientation is an $H_\infty$ map, Amer. J. Math. 126 (2004), 247–334.
  • A. P. Bahri and M. E. Mahowald, A direct summand in $H^{\ast} (M\mathrm{O}\langle8\rangle, Z_{2})$, Proc. Amer. Math. Soc. 78 (1980), 295–298.
  • A. Baker and A. Lazarev, On the Adams spectral sequence for $R$-modules, Algebr. Geom. Topol. 1 (2001), 173–199 (electronic).
  • R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029.
  • R. R. Bruner, $\mathrm{Ext}$ in the nineties, Algebraic topology (Oaxtepec, 1991), Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 71–90.
  • H. Cartan, Sur les groupes d'Eilenberg–Mac Lane. II, Proc. Nat. Acad. Sci. USA 40 (1954), 704–707.
  • D.-E. Diaconescu, G. Moore and E. Witten, $E_8$ gauge theory, and a derivation of $K$-theory from M-theory, Adv. Theor. Math. Phys. 6 (2002), 1031–1134 (2003).
  • S. R. Edwards, On the spin bordism of $B(E_8\times E_8)$, Illinois J. Math. 35 (1991), 683–689.
  • J. N. K. Francis, Spin bordism of $B\mathit{Spin}$ and $K({Z},4)$: Integrality and index theory, Undergraduate thesis.
  • M. A. Hill, The 3-local $\mathrm{tmf}$-homology of $B\Sigma _3$, Proc. Amer. Math. Soc. 135 (2007), 4075–4086 (electronic).
  • M. J. Hopkins, Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, vol. I (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 291–317.
  • M. J. Hopkins and M. Mahowald, From elliptic curves to homotopy theory, available on the Hopf Archive, 1998.
  • M. J. Hopkins and H. R. Miller, Elliptic curves and stable homotopy I, in preparation.
  • M. A. Hovey and D. C. Ravenel, The $7$-connected cobordism ring at $p=3$, Trans. Amer. Math. Soc. 347 (1995), 3473–3502.
  • P. S. Landweber and R. E. Stong, A bilinear form for Spin manifolds, Trans. Amer. Math. Soc. 300 (1987), 625–640.
  • D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Orlando, FL, 1986.
  • C. Rezk, Supplimentary notes for math 512, available at
  • J.-P. Serre, Sur les groupes d'Eilenberg–MacLane, C. R. Acad. Sci. Paris 234 (1952), 1243–1245.
  • R. E. Stong, Appendix: Calculation of $\Omega^{\mathrm{Spin}}_ {11}(K(Z,4))$, Workshop on unified string theories (Santa Barbara, Calif., 1985), World Scientific, Singapore, 1986, pp. 430–437.
  • R. E. Stong, Determination of $H^{\ast} (\mathrm{BO}(k,\ldots,\infty ),Z_{2})$ and $H^{\ast} (\mathrm{BU}(k,\ldots,\infty),Z_{2})$, Trans. Amer. Math. Soc. 107 (1963), 526–544.