Gysin Map and Atiyah-Hirzebruch Spectral Sequence

Ferrari Ruffino, Fabio

Bollettino dell'Unione Matematica Italiana, Tome 4 (2011), p. 263-273 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Résumé

We discuss the relations between the Atiyah-Hirzebruch spectral sequence and the Gysin map for a multiplicative cohomology theory, on spaces having the homotopy type of a finite CW-complex. In particular, let us fix such a multiplicative cohomology theory $h^{*}$ and let us consider a smooth manifold $X$ of dimension $n$ and a compact submanifold $Y$ of dimension $p$ , satisfying suitable hypotheses about orientability. We prove that, starting the Atiyah-Hirzebruch spectral sequence with the Poincaré dual of $Y$ in $X$ , which, in our setting, is a simplicial cohomology class with coefficients in $h^{0}\{*\}$ , if such a class survives until the last step, it is represented in $E^{n-p,0}_{\infty}$ by the image via the Gysin map of the unit cohomology class of $Y$ . We then prove the analogous statement for a generic cohomology class on $Y$ .

Publié le : 2011-06-01

MR 2840607 | Zbl 1241.55011

@article{BUMI_2011_9_4_2_263_0,
     author = {Fabio Ferrari Ruffino},
     title = {Gysin Map and Atiyah-Hirzebruch Spectral Sequence},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {4},
     year = {2011},
     pages = {263-273},
     zbl = {1241.55011},
     mrnumber = {2840607},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2011_9_4_2_263_0}
}

Ferrari Ruffino, Fabio. Gysin Map and Atiyah-Hirzebruch Spectral Sequence. Bollettino dell'Unione Matematica Italiana, Tome 4 (2011) pp. 263-273. http://gdmltest.u-ga.fr/item/BUMI_2011_9_4_2_263_0/

Bibliographie

[1] Atiyah, M. - Hirzebruch, F., Vector Bundles and Homogeneous Spaces, Michael Atiyah: Collected works, v. 2. | Zbl 0108.17705

[2] Bohr, C. - Hanke, B. - Kotschick, D., Cycles, submanifolds and structures on normal bundles, Manuscripta Math., 108 (2002), 483-494. | MR 1923535 | Zbl 1009.57043

[3] Bredon, G. E., Topology and geometry, Springer-Verlag, 1993. | MR 1224675

[4] Cartan, H. - Eilenberg, S., Homological algebra, Princeton University Press, 1956. | MR 77480

[5] Dold, A., Relations between ordinary and extraordinary homology, Colloquium on Algebraic Topology, Institute of Mathematics Aarhus University (1962), 2-9.

[6] Ferrari Ruffino, F. - Savelli, R., Comparing two approaches to the K-theory classification of D-branes, Journal of Geometry and Physics, 61 (2011), 191-212. | MR 2746991 | Zbl 1207.81134

[7] Griffiths, P. - Harris, J., Principles of algebraic geometry, John Wiley Sons, 1978. | MR 507725 | Zbl 0408.14001

[8] Munkres, J. R., Elementary Differential Topology, Princeton University Press, 1968. | MR 198479

[9] Rudyak, Y. B., On Thom spectra, orientability and cobordism, Springer monographs in mathematics. | MR 1627486 | Zbl 0906.55001