# Characteristic classes as complete obstructions

## Abstract

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group.

## Keywords

Characteristic class Group reduction Obstruction Principal bundle Classifying space## Notes

### Acknowledgements

This paper is part of my PhD thesis, and I would like to thank my advisor Kathryn Hess for her support and for her constructive feedback on earlier versions of this paper. I am grateful to Jeffrey Carlson for suggesting an alternative description of the plus-cohomology groups, and to Viktoriya Ozornova for many extremely helpful discussions. The quality of exposition benefited from the referee’s comments.

## References

- 1.Cencelj, M., Kosta, N.M., Vavpetič, A., et al.: \( G \)-complexes with a compatible CW structure. J. Math. Kyoto Univ.
**43**(3), 585–597 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Cohen, R.: The topology of fiber bundles, online notes (1998). https://www.researchgate.net/publication/242606546_The_Topology_of_Fiber_Bundles_Lecture_Notes
- 3.Dold, A.: Partitions of unity in the theory of fibrations. Ann. Math. (2)
**78**, 223–255 (1963)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Earle, C.J., Eells, J.: The diffeomorphism group of a compact riemann surface. Bull. Am. Math. Soc.
**73**(4), 557–559 (1967)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Farjoun, E.D.: Cellular spaces, null spaces and homotopy localization. In: Lecture Notes in Mathematics, vol. 1622. Springer, Berlin (1996)Google Scholar
- 6.Farjoun, E.D., Hess, K.: Normal and conormal maps in homotopy theory. Homol. Homotopy Appl.
**14**(1), 79–112 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Fritsch, R., Piccinini, R.A.: Cellular Structures in Topology, Cambridge Studies in Advanced Mathematics, vol. 19. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
- 8.Gabai, D.: The smale conjecture for hyperbolic 3-manifolds. J. Differ. Geom.
**58**(1), 113–149 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Hatcher, A.: On the diffeomorphism group of \(S^1 \times S^2\). Proc. Am. Math. Soc.
**83**(2), 427–430 (1981)zbMATHGoogle Scholar - 10.Hatcher, A.: A proof of the Smale conjecture, \({\rm Diff}(S^{3})\simeq {\rm O}(4)\). Ann. Math. (2)
**117**(3), 553–607(1983)Google Scholar - 11.Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
- 12.Illman, S.: The equivariant triangulation theorem for actions of compact lie groups. Math. Ann.
**262**(4), 487–501 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Lundell, A.T., Weingram, S.: The Topology of CW Complexes, The University Series in Higher Mathematics. Van Nostrand Reinhold Co., New York (1969)CrossRefzbMATHGoogle Scholar
- 14.May, J.P. et al.: Equivariant homotopy and cohomology theory, volume 91 of CBMS regional conference series in mathematics. In: Published for the Conference Board of the Mathematical Sciences, Washington, DC, p. 88 (1996)Google Scholar
- 15.May, J.P.: A concise course in algebraic topology. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1999)Google Scholar
- 16.Milnor, J.: Construction of universal bundles. I. Ann. Math. (2)
**63**, 272–284 (1956)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Milnor, J.: Construction of universal bundles II. Ann. Math. (2)
**63**, 430–436 (1956)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Milnor, J.: On spaces having the homotopy type of a \({\rm CW}\)-complex. Trans. Am. Math. Soc.
**90**, 272–280 (1959)MathSciNetzbMATHGoogle Scholar - 19.Milgram, R.J.: The bar construction and abelian \(H\)-spaces. Ill. J. Math.
**11**, 242–250 (1967)MathSciNetzbMATHGoogle Scholar - 20.Mitchell, S.: Notes on principal bundles and classifying spaces, online notes (2011). https://sites.math.washington.edu/~mitchell/Atopc/prin.pdf
- 21.Mac Lane, S.: Categories for the working mathematician, 2 ed. In: Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)Google Scholar
- 22.Milnor, J.W., Stasheff, J.D.: Characteristic classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974, Annals of Mathematics Studies, No. 76Google Scholar
- 23.Munkres, J.R.: Topology: a First Course. Prentice-Hall Inc., Englewood Cliffs (1975)zbMATHGoogle Scholar
- 24.Nikolaus, T., Schreiber, U., Stevenson, D.: Principal \(\infty \)-bundles: general theory. J. Homotopy Relat. Struct.
**10**(4), 749–801 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 25.Piccinini, R.A.: Lectures on homotopy theory, North-Holland Mathematics Studies, vol. 171. North-Holland Publishing Co., Amsterdam (1992)Google Scholar
- 26.Rovelli, M.: A looping-delooping adjunction for topological spaces. Homol. Homotopy Appl.
**19**(1), 37–57 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Rovelli, M.: Towards new invariants for principal bundles, Ph.D. thesis (2017). https://infoscience.epfl.ch/record/226464
- 28.Segal, G.: Categories and cohomology theories. Topology
**13**, 293–312 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Smale, S.: Diffeomorphisms of the 2-sphere. Proc. Am. Math. Soc.
**10**(4), 621–626 (1959)MathSciNetzbMATHGoogle Scholar - 30.Sati, H., Schreiber, U., Stasheff, J.: Fivebrane structures. Rev. Math. Phys.
**21**(10), 1197–1240 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Steenrod, N.E.: Cohomology operations. In: Lectures by N. E. Steenrod Written and Revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50. Princeton University Press, Princeton (1962)Google Scholar
- 32.Steenrod, N.: The Topology of Fibre Bundles, Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999, Reprint of the 1957 edition, Princeton PaperbacksGoogle Scholar
- 33.Strickland, N.: The category of CGWH spaces, online notes (2009). http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf
- 34.Whitehead, G.W.: Fiber spaces and the Eilenberg homology groups. Proc. Nat. Acad. Sci. USA
**38**, 426–430 (1952)MathSciNetCrossRefzbMATHGoogle Scholar