Gysin Map and Atiyah-Hirzebruch Spectral Sequence

Fabio Ferrari Ruffino

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 2, page 263-273
  • ISSN: 0392-4041

Abstract

top
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 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 .

How to cite

top

Ferrari Ruffino, Fabio. "Gysin Map and Atiyah-Hirzebruch Spectral Sequence." Bollettino dell'Unione Matematica Italiana 4.2 (2011): 263-273. <http://eudml.org/doc/290742>.

@article{FerrariRuffino2011,
abstract = {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 $Y$.},
author = {Ferrari Ruffino, Fabio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {263-273},
publisher = {Unione Matematica Italiana},
title = {Gysin Map and Atiyah-Hirzebruch Spectral Sequence},
url = {http://eudml.org/doc/290742},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Ferrari Ruffino, Fabio
TI - Gysin Map and Atiyah-Hirzebruch Spectral Sequence
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/6//
PB - Unione Matematica Italiana
VL - 4
IS - 2
SP - 263
EP - 273
AB - 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 $Y$.
LA - eng
UR - http://eudml.org/doc/290742
ER -

References

top
  1. ATIYAH, M. - HIRZEBRUCH, F., Vector Bundles and Homogeneous Spaces, Michael Atiyah: Collected works, v. 2. Zbl0108.17705
  2. BOHR, C. - HANKE, B. - KOTSCHICK, D., Cycles, submanifolds and structures on normal bundles, Manuscripta Math., 108 (2002), 483-494. Zbl1009.57043MR1923535DOI10.1007/s002290200279
  3. BREDON, G. E., Topology and geometry, Springer-Verlag, 1993. MR1224675DOI10.1007/978-1-4757-6848-0
  4. CARTAN, H. - EILENBERG, S., Homological algebra, Princeton University Press, 1956. MR77480
  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. Zbl1207.81134MR2746991DOI10.1016/j.geomphys.2010.10.001
  7. GRIFFITHS, P. - HARRIS, J., Principles of algebraic geometry, John Wiley Sons, 1978. Zbl0408.14001MR507725
  8. MUNKRES, J. R., Elementary Differential Topology, Princeton University Press, 1968. MR198479
  9. RUDYAK, Y. B., On Thom spectra, orientability and cobordism, Springer monographs in mathematics. Zbl0906.55001MR1627486

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.