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Keywords:
Chern class; characteristic rank; cup length; chern rank
Chern class; characteristic rank; cup length; chern rank
Summary:
Motivated by the work of A.\,C. Naolekar and A.\,S. Thakur (2014) we introduce notions of upper chern rank and even cup length of a finite connected CW-complex and prove that upper chern rank is a homotopy invariant. It turns out that determination of upper chern rank of a space $X$ sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over $X$ or not. For a closed connected $d$-dimensional complex manifold we obtain an upper bound of its even cup length. For a finite connected even dimensional CW-complex with its upper chern rank equal to its dimension, we provide a method of computing its even cup length. Finally, we compute upper chern rank of many interesting spaces.
References:
[1] Adams J. F.: Vector fields on spheres. Ann. of Math. (2) 75 (1962), no. 3, 603–632. DOI 10.2307/1970213 | MR 0139178
[2] Husemoller D.: Fibre Bundles. McGraw-Hill Book Co., New York, 1966. MR 0229247
[3] Korbaš J.: The cup length of oriented Grassmannians vs a new bound for zero-cobordant manifolds. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 1, 69–81. DOI 10.36045/bbms/1267798499 | MR 2656672
[4] McCleary J.: A User's Guide to Spectral Sequences. Cambridge Studies in Advanced Mathematics, 58, Cambridge University Press, Cambridge, 2001. MR 1793722 | Zbl 0959.55001
[5] Milnor J., Stasheff J.: Characteristic Classes. Annals of Mathematics Studies, 76, Princeton University Press, Princeton, University of Tokyo Press, Tokyo, 1974. MR 0440554 | Zbl 1079.57504
[6] Naolekar A. C.: Realizing cohomology classes as Euler classes. Math. Slovaca 62 (2012), no. 5, 949–966. DOI 10.2478/s12175-012-0057-2 | MR 2981832
[7] Naolekar A. C., Thakur A. S.: Note on the characteristic rank of vector bundles. Math. Slovaca 64 (2014), no. 6, 1525–1540. DOI 10.2478/s12175-014-0289-4 | MR 3298036
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