Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T00:29:25.126Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Berstein–Svarc theorem in dimension 2

Published online by Cambridge University Press:  01 March 2009

ALEXANDER N. DRANISHNIKOV  and
YULI B. RUDYAK
ALEXANDER N. DRANISHNIKOV
Affiliation:
Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611, U.S.A. e-mail: [email protected]; [email protected]
YULI B. RUDYAK
Affiliation:
Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611, U.S.A. e-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that for any group π with cohomological dimension at least n the nth power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n:

Theorem. For a connected complex X with dim X = cat X = n, we have ≠ 0 whereis the Berstein class of X.

Previously it was known for n ≥ 3.

We also prove that, for every map f: M → N of degree ±1 of closed orientable manifolds, the fundamental group of N is free provided that the fundamental group of M is.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 407 - 413
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Berstein, I.On the Lusternik–Schnirelmann category of Grassmannians. Math. Proc. Camb. Phil. Soc. 79 404 (1976), 129–134.CrossRefGoogle Scholar
[2]Brown, K.Cohomology of groups. Graduate Texts in Mathematics, 87 (Springer, 1994.)Google Scholar
[3]Bredon, G.Sheaf theory. Graduate Texts in Mathematics, 170 (Springer, 1997).CrossRefGoogle Scholar
[4]Cornea, O., Lupton, G., Oprea, J. and Tanré, D.Lusternik–Schnirelmann category. Mathematical Surveys and Monographs, 103 (Ameri. Math. Soc., 2003.)CrossRefGoogle Scholar
[5]Eilenberg, S. and Ganea, T.On the Lusternik–Schnirelmann category of abstract groups. Ann. of Math. (2) 65 (1957), 517–518.CrossRefGoogle Scholar
[6]Oprea, J. and Rudyak, Y.Detecting elements and Lusternik–Schnirelmann category of 3-manifolds. Lusternik–Schnirelmann category and related topics (South Hadley, MA, 2001), 181–191, Contemp. Math. 316 (Amer. Math. Soc., 2002).CrossRefGoogle Scholar
[7]Stallings, J.Groups of dimension 1 are locally free. Bull. Amer. Math. Soc. 74 (1968), 361–364.CrossRefGoogle Scholar
[8]Švarc, A.The genus of a fibered space. Trudy Moskov. Mat. Obšč 10, 11 (1961 and 1962), 217–272, 99–126, (in Amer. Math. Soc. Transl. Series 2, vol 55 (1966)).Google Scholar
[9]Swan, R.Groups of cohomological dimension one. J. Algebra 12 (1969), 585–610.CrossRefGoogle Scholar