ON THE SYMMETRIC SQUARES OF COMPLEX AND QUATERNIONIC PROJECTIVE SPACE

BOOTE, YUMI ; RAY, NIGEL

Glasgow mathematical journal, Tome 60 (2018) no. 3, pp. 703-729

Résumé

The problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. We offer a solution for the complex and quaternionic projective spaces $\mathbb{K}$Pn, by utilising their rich geometrical structure. Our description involves generators and relations, and our methods entail ideas from the literature of quantum chemistry, theoretical physics, and combinatorics. We begin with the case $\mathbb{K}$P∞, and then identify the truncation required for passage to finite n. The calculations rely upon a ladder of long exact cohomology sequences, which compares cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. These incorporate the one-point compactifications of classic configuration spaces of unordered pairs of points in $\mathbb{K}$Pn, which are identified as Thom spaces by combining Löwdin's symmetric orthogonalisation (and its quaternionic analogue) with a dash of Pin geometry. The relations in the ensuing cohomology rings are conveniently expressed using generalised Fibonacci polynomials. Our conclusions are compatible with those of Gugnin mod torsion and Nakaoka mod 2, and with homological results of Milgram.

Détail

DOI : 10.1017/S0017089518000034

Bibliographie

[1] 1. Adem, A., Leida, J. and Ruan, Y., Orbifolds and stringy topology, Cambridge tracts in mathematics, vol. 171 (Cambridge University Press, Cambridge, UK, 2007). Google Scholar

[2] 2. Aguilar, M. A. and Prieto, C., A classification of cohomology transfers for ramified covering maps, Fund. Math. 189 (1) (2006), 1–25. Google Scholar

[3] 3. Amdeberhan, T., Chen, X., Moll, V. and Sagan, B., Generalized Fibonacci polynomials and Fibonomial coefficients, Ann. Comb. 18 (4) (2014), 541–562. Google Scholar

[4] 4. Bakuradze, M., The transfer and symplectic cobordism, Trans. Am. Math. Soc. 349 (11) (1997), 4385–4399. Google Scholar

[5] 5. Berg, M., Dewitt-Morette, C., Gwo, S. and Kramer, E., The Pin groups in physics: C, P, and T, Rev. Math. Phys. 13 (08) (2001), 953–1034. Google Scholar

[6] 6. Boote, Y., On the symmetric square of quaternionic projective space, PhD Thesis, (University of Manchester, 2016). Google Scholar

[7] 7. Boote, Y. and Ray, N., On the symmetric squares of complex and quaternionic projective space, arXiv:1603.02066v2 [math.AT] (2016). Google Scholar

[8] 8. Boote, Y. and Ray, N., Compactifications of configuration spaces of pairs; even spaces and the octonionic projective plane, in preparation. Google Scholar

[9] 9. Borisov, L., Halpern, M. B. and Schweigert, C., Systematic approach to cyclic orbifolds, Int. J. Modern Phys. A 13 (1998), 125–168. Google Scholar | DOI

[10] 10. Bredon, G. E., Introduction to compact transformation groups, Pure and applied mathematics, vol. 46 (Academic Press, New York and London, 1972). Google Scholar

[11] 11. Cadek, M., The cohomology of BO(n) with twisted integer coefficients, J. Math. Kyoto Univ. 39 (2) (1999), 277–286. Google Scholar

[12] 12. Cartan, H. and Eilenberg, S., Homological algebra, Princeton mathematical series, vol. 19 (Princeton University Press, Princeton, New Jersey, 1956). Google Scholar

[13] 13. Dold, A., Lectures on algebraic topology, Classics in mathematics (Springer-Verlag, Berlin, Heidelberg, 1995). Google Scholar | DOI

[14] 14. Dold, A. and Thom, R., Quasifaserungen und unendliche symmetrische produkte, Ann. Math. 67 (2) (1958), 239–281. Google Scholar

[15] 15. Dominguez, C., Gonzalez, J., and Landweber, P., The integral cohomology of configuration spaces of pairs of points in real projective spaces, Forum Math. 25 (2013), 1217–1248. Google Scholar

[16] 16. Feder, S., The reduced symmetric product of projective spaces and the generalized Whitney Theorem, Illinois J. Math. 16 (2) (1972), 323–329. Google Scholar

[17] 17. Gould, H. W., A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart. 19 (3) (1981), 250–257. Google Scholar

[18] 18. Gugnin, D. V.. On the integral cohomology ring of symmetric products. arXiv:1502.01862v3 [math.AT] (2015). Google Scholar

[19] 19. Hambleton, I., Kreck, M. and Teichner, P., Nonorientable 4-manifolds with fundamental group of order 2, Trans. Am. Math. Soc. 344 (2) (1994), 649–665. Google Scholar

[20] 20. Husemoller, D., Fiber bundles, 3rd ed. Graduate texts in mathematics (Springer-Verlag, New York, 1994). Google Scholar

[21] 21. Jacobsthal, E., Fibonaccische Polynome und Kreisteilungsgleichungen, Sitzungsberichte Berl. Math. Ges. 17 (1919–1920), 43–57. Google Scholar

[22] 22. James, I., Thomas, E., Toda, H. and Whitehead, G. W., The symmetric square of a sphere, J. Math. Mech 12 (5)(1963), 771–776. Google Scholar

[23] 23. Kallel, S., Symmetric products, duality and homological dimension of configuration spaces, Geom. Topol. Monogr. 13 (2008), 499–527. Google Scholar | DOI

[24] 24. Kallel, S. and Karoui, R., Symmetric joins and weighted barycenters, Adv. Nonlinear Stud. 11 (2011), 117–143. Google Scholar

[25] 25. Kallel, S. and Taamallah, W., The geometry and fundamental group of permutation products and fat diagonals, Canad. J. Math. 65 (3) (2013), 575–599. Google Scholar

[26] 26. Kirby, R. and Taylor, L., Pin structures on low dimensional manifolds, Geometry of low dimensional manifolds 2 (Durham, 1989), London Mathematical Society lecture note series, vol. 151 (Cambridge University Press, Cambridge, UK, 1990), 177–242. Google Scholar

[27] 27. Löwdin, P.-O., On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals, J. Chem. Phys. 18 (3) (1950), 365–375. Google Scholar

[28] 28. Milgram, R. J., The homology of symmetric products, Trans. Am. Math. Soc. 138 (1969), 251–265. Google Scholar

[29] 29. Mimura, M. and Toda, H., Topology of Lie groups I and II, Translations of mathematical monographs, vol. 91 (American Mathematical Society, Providence, Rhode Island, 1991). Google Scholar

[30] 30. Mitchell, S. A., Notes on principal bundles and classifying spaces, http://www.math.washington.edu/mitchell/Notes/prin.pdf (June 2011). Google Scholar

[31] 31. Mitchell, S. A. and Priddy, S. B., Symmetric product spectra and splittings of classifying spaces, Am. J. Math. 106 (1) (1984), 219–232. Google Scholar

[32] 32. Morse, M., The calculus of variations in the large, American Mathematical Society, vol. 18 (Colloquium Publications, 1934). Current edition, 2012. Google Scholar | DOI

[33] 33. Nakaoka, M., Cohomology of the p-fold cyclic products, Proc. Japan Acad. 31 (10) (1955), 665–669. Google Scholar

[34] 34. Nakaoka, M., Cohomology theory of a complex with a transformation of prime period and its applications, J. Inst. Poly. Osaka City Univ. Ser. A, Math. 7 (1–2) (1956) 51–102. Google Scholar

[35] 35. Nakaoka, M., Decomposition theorem for homology groups of symmetric groups, Ann. Math. 71 (1) (1960), 16–42. Google Scholar

[36] 36. Porteous, I. R., Clifford algebras and the classical groups, Cambridge studies in advanced mathematics, volume 50 (Cambridge University Press, Cambridge, UK, 1995). Google Scholar

[37] 37. Roush, F. W., On some torsion classes in symplectic cobordism, preprint (1972). Google Scholar

[38] 38. Smith, L., Transfer and ramified coverings, Math. Proc. Cambridge Philos. Soc. 93 (1983), 485–493. Google Scholar

[39] 39. Steenrod, N. E. and Epstein, D. B. A., Cohomology operations, Annals of Mathematics studies, vol. 50 (Princeton University Press, Princeton, New Jersey, 1962). Google Scholar

[40] 40. Totaro, B., The integral cohomology of the Hilbert scheme of two points, Forum Math. Sigma 4 (2016), e8, 22pp. Google Scholar

[41] 41. Yasui, T., The reduced symmetric product of a complex projective space and the embedding problem. Hiroshima Math. J. 1 (1971), 27–40. Google Scholar

[42] 42. Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997), 21–57. Google Scholar