On manifolds homotopy equivalent to the total spaces of S 7 -bundles over S 8

Ajay Raj; Tibor Macko

Archivum Mathematicum (2024)

  • Volume: 060, Issue: 3, page 125-134
  • ISSN: 0044-8753

Abstract

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We calculate the structure sets in the sense of surgery theory of total spaces of bundles over eight-dimensional sphere with fibre a seven-dimensional sphere, in which manifolds homotopy equivalent to the total spaces are organized, and we investigate the question, which of the elements in these structure sets can be realized as such bundles.

How to cite

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Raj, Ajay, and Macko, Tibor. "On manifolds homotopy equivalent to the total spaces of $S^7$-bundles over $S^8$." Archivum Mathematicum 060.3 (2024): 125-134. <http://eudml.org/doc/299435>.

@article{Raj2024,
abstract = {We calculate the structure sets in the sense of surgery theory of total spaces of bundles over eight-dimensional sphere with fibre a seven-dimensional sphere, in which manifolds homotopy equivalent to the total spaces are organized, and we investigate the question, which of the elements in these structure sets can be realized as such bundles.},
author = {Raj, Ajay, Macko, Tibor},
journal = {Archivum Mathematicum},
keywords = {vector bundle; sphere bundle over sphere; microbundle; homotopy equivalence; homeomorphism; surgery; characteristic class},
language = {eng},
number = {3},
pages = {125-134},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On manifolds homotopy equivalent to the total spaces of $S^7$-bundles over $S^8$},
url = {http://eudml.org/doc/299435},
volume = {060},
year = {2024},
}

TY - JOUR
AU - Raj, Ajay
AU - Macko, Tibor
TI - On manifolds homotopy equivalent to the total spaces of $S^7$-bundles over $S^8$
JO - Archivum Mathematicum
PY - 2024
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 060
IS - 3
SP - 125
EP - 134
AB - We calculate the structure sets in the sense of surgery theory of total spaces of bundles over eight-dimensional sphere with fibre a seven-dimensional sphere, in which manifolds homotopy equivalent to the total spaces are organized, and we investigate the question, which of the elements in these structure sets can be realized as such bundles.
LA - eng
KW - vector bundle; sphere bundle over sphere; microbundle; homotopy equivalence; homeomorphism; surgery; characteristic class
UR - http://eudml.org/doc/299435
ER -

References

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  1. Adams, J.F., 10.1016/0040-9383(66)90004-8, Topology 5 (1966), 21–71. (1966) MR0198470DOI10.1016/0040-9383(66)90004-8
  2. Bredon, G.E., Topology and Geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag Inc., New York, 1993. (1993) MR1224675
  3. Chang, S., Weinberger, S., A course on surgery theory, Annals of Mathematics Studies, vol. 211, Princeton University Press, 2021. (2021) MR4207574
  4. Crowley, D., Escher, C.M., 10.1016/S0926-2245(03)00012-3, Differential Geom. Appl. 18 (3) (2003), 363–380. (2003) MR1975035DOI10.1016/S0926-2245(03)00012-3
  5. Dold, A., Lashof, R., 10.1215/ijm/1255455128, Illinois J. Math. 3 (1959), 285–305. (1959) MR0101521DOI10.1215/ijm/1255455128
  6. Grey, M., On the classification of total space of S 7 -bundles over S 8 , Master's thesis, Humboldt University, Berlin, 2012. (2012) 
  7. Kirby, R.C., Siebenmann, L.C., Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, vol. 88, Princeton University Press, 1977. (1977) MR0645390
  8. Kitchloo, N., Shankar, K., 10.1155/S1073792801000186, Int. Math. Res. Not. IMRN 2001 (8) (2001), 381–394. (2001) MR1827083DOI10.1155/S1073792801000186
  9. Madsen, I.B., Milgram, R.J., The Classifying Spaces For Surgery and Cobordism of Manifolds, Princeton University Press, New Jersey, 1979. (1979) MR0548575
  10. Milnor, J., 10.2307/1969983, Ann. of Math. (2) 64 (2) (1956), 399–405. (1956) MR0082103DOI10.2307/1969983
  11. Ranicki, A., Algebraic L -theory and topological manifolds, Cambridge University Press, 1992. (1992) MR1211640
  12. Ranicki, A., On The Novikov Conjecture, Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 272–337. (1995) MR1388304
  13. Ranicki, A., Algebraic and Geometric Surgery, Oxford Mathematical Monograph, Oxford Science Publication, 2002, ISBN 978-0198509240. (2002) MR2061749
  14. Shimada, N., 10.1017/S0027763000021942, Nagoya Math. J. 12 (1957), 59–69. (1957) Zbl0145.20303MR0096223DOI10.1017/S0027763000021942
  15. Tamura, I., 10.2969/jmsj/00920250, J. Math. Soc. Japan 9 (2) (1957), 250–262. (1957) MR0091475DOI10.2969/jmsj/00920250
  16. Tamura, I., 10.2969/jmsj/01010029, J. Math. Soc. Japan 10 (1) (1958), 29–42. (1958) MR0096225DOI10.2969/jmsj/01010029
  17. Toda, H., Saito, Y., Yokota, I., Notes on the generator of π 7 ( S O ( n ) ) , Mem. Coll. Sci. Univ. Kyoto 30 (1957), 227–230. (1957) MR0091474
  18. Wall, C.T.C., Surgery on Compact Manifolds, Mathematical Surveys and Monographs, vol. 69, AMS, 1999. (1999) MR1687388
  19. Whitehead, G.W., 10.2307/1969085, Ann. of Math. (2) 40 (3) (1946), 460–475. (1946) MR0016672DOI10.2307/1969085
  20. Whitehead, G.W., 10.2307/1969506, Ann. of Math. (2) 50 (1) (1950), 192–237. (1950) MR0041435DOI10.2307/1969506

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