TY - JOUR
T1 - On the topological complexity of aspherical spaces
AU - Farber, Michael
AU - Mescher, Stephan
N1 - Publisher Copyright:
© 2020 World Scientific Publishing Company.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517-518] expresses the Lusternik-Schnirelmann category of an Eilenberg-MacLane space K(?, 1) as the cohomological dimension of the group ?. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg-MacLane spaces K(?, 1). One of our main results states that in the case when the group ? is hyperbolic in the sense of Gromov, the topological complexity TC(K(?, 1)) either equals or is by one larger than the cohomological dimension of ? × ?. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.
AB - The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517-518] expresses the Lusternik-Schnirelmann category of an Eilenberg-MacLane space K(?, 1) as the cohomological dimension of the group ?. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg-MacLane spaces K(?, 1). One of our main results states that in the case when the group ? is hyperbolic in the sense of Gromov, the topological complexity TC(K(?, 1)) either equals or is by one larger than the cohomological dimension of ? × ?. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.
KW - Topological complexity
KW - aspherical spaces
KW - topological robotics
UR - http://www.scopus.com/inward/record.url?scp=85053764250&partnerID=8YFLogxK
U2 - 10.1142/S1793525319500511
DO - 10.1142/S1793525319500511
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AN - SCOPUS:85053764250
SN - 1793-5253
VL - 12
SP - 293
EP - 319
JO - Journal of Topology and Analysis
JF - Journal of Topology and Analysis
IS - 2
ER -