Computation of Nielsen and Reidemeister coincidence numbers for multiple maps
Keywords
Topological coincidence theory, Nielsen coincidence number, nilmanifoldsAbstract
Let $f_1,\ldots,f_k\colon M\to N$ be maps between closed manifolds, $N(f_1,\ldots,f_k)$ and $R(f_1,\ldots,f_k)$ be the Nielsen and the Reideimeister coincidence numbers, respectively. In this note, we relate $R(f_1,\ldots,f_k)$ with $R(f_1,f_2),\ldots,R(f_1,f_k)$. When $N$ is a torus or a nilmanifold, we compute $R(f_1,\ldots,f_k)$ which, in these cases, is equal to $N(f_1,\ldots,f_k)$.References
V. del Barco, Symplectic structures on nilmanifolds: an obstruction for their existence, J. Lie Theory 24 (2014), no. 3, 889–908.
C. Benson and C. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513–518.
C. Biasi, A.K.M. Libardi and T.F.M. Monis, The Lefschetz coincidence class of p-maps, Forum Math. 27 (2015), no. 3, 1717–1728.
R. Brooks, On the sharpness of the ∆2 and ∆1 Nielsen numbers, J. Reine Angew. Math. 259 (1973), 101–108.
R. Brooks, Certain subgroups of the fundamental group and the number of roots of f (x) = a, Amer. J. Math. 95 (1973), no. 4, 720–728.
R. Brooks, On removing coincidences of two maps when only one, rather than both of them, may be deformed by a homotopy, Pacific J. Math. 40 (1972), 45–52.
D. Gonçalves and P. Wong, Coincidence Wecken property for nilmanifolds, Acta. Math. Sin. (English Ser.) 35 (2019), no. 2, 239–244.
D. Gonçalves and P. Wong, Obstruction theory and coincidences of maps between nilmanifolds, Archiv Math. 84 (2005), 568–576.
D. Gonçalves and P. Wong, Nilmanifolds are Jiang-type spaces for coincidences, Forum Math. 13 (2001), 133–141.
D. Gonçalves and P. Wong, Wecken property for roots, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2779–2782.
D. Gonçalves and P. Wong, Homogeneous spaces in coincidence theory II, Forum Math. 17 (2005), 297–313.
P. Heath, Groupoids and relations among Reidemeister and among Nielsen numbers, Topology Appl. 181 (2015), 3–33.
T.F.M. Monis and S. Spież, Lefschetz coincidence class for several maps, J. Fixed Point Theory Appl. 18 (2016), no. 1, 61–76.
T.F.M. Monis and P. Wong, Obstruction theory for coincidences of multiple maps, Topology Appl. 229 (2017), 213–225.
S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311–333.
P.C. Staecker, Nielsen equalizer theory, Topology Appl. 158 (2011), no. 13 , 1615–1625.
P. Wong, Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds, J. Reine Angew. Math. 524 (2000), 185–204.
P. Wong, Coincidence theory for spaces which fiber over a nilmanifold, Fixed Point Theory Appl. (2004), no. 2, 89–95.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
