TY - JOUR
T1 - Homology of planar polygon spaces
AU - Farber, M.
AU - Schütz, D.
N1 - Funding Information:
Acknowledgements M. Farber was supported by the Royal Society Wolfson Research Merit Award and by EPSRC grant “Applications of Algebraic Topology”.
PY - 2007/3
Y1 - 2007/3
N2 - In this paper, we study topology of the variety of closed planar n-gons with given side lengths l1,...,ln. The moduli space M l where l =(l1,...,ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M l as functions of the length vector l = (l1,...,l n). We also find sharp upper bounds on the sum of Betti numbers of Ml depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.
AB - In this paper, we study topology of the variety of closed planar n-gons with given side lengths l1,...,ln. The moduli space M l where l =(l1,...,ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M l as functions of the length vector l = (l1,...,l n). We also find sharp upper bounds on the sum of Betti numbers of Ml depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.
KW - Morse theory of manifolds with involutions
KW - Polygon spaces
KW - Varieties of linkages
UR - http://www.scopus.com/inward/record.url?scp=34547150256&partnerID=8YFLogxK
U2 - 10.1007/s10711-007-9139-7
DO - 10.1007/s10711-007-9139-7
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AN - SCOPUS:34547150256
SN - 0046-5755
VL - 125
SP - 75
EP - 92
JO - Geometriae Dedicata
JF - Geometriae Dedicata
IS - 1
ER -