TY - JOUR
T1 - Closed 1-forms in topology and geometric group theory
AU - Farber, M.
AU - Geoghegan, R.
AU - Schütz, D.
PY - 2010
Y1 - 2010
N2 - In this article we describe relations of the topology of closed 1-forms to the group-theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW-complexes and show that many properties of the group-theoretic version have analogous statements. In particular, we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik-Schnirelmann category of a closed 1-form and to the existence of a non-singular closed 1-form in a given cohomology class on a high-dimensional closed manifold. Bibliography: 32 titles.
AB - In this article we describe relations of the topology of closed 1-forms to the group-theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW-complexes and show that many properties of the group-theoretic version have analogous statements. In particular, we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik-Schnirelmann category of a closed 1-form and to the existence of a non-singular closed 1-form in a given cohomology class on a high-dimensional closed manifold. Bibliography: 32 titles.
KW - Lusternik-Schnirelmann category
KW - Movability of homology classes
KW - Novikov ring
KW - Sigma invariants
UR - http://www.scopus.com/inward/record.url?scp=77954804069&partnerID=8YFLogxK
U2 - 10.1070/RM2010v065n01ABEH004663
DO - 10.1070/RM2010v065n01ABEH004663
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:77954804069
VL - 65
SP - 143
EP - 172
JO - Russian Mathematical Surveys
JF - Russian Mathematical Surveys
SN - 0036-0279
IS - 1
ER -