Nielsen root theory and Hopf degree theory

The Nielsen root number N(f ; c) of a map f : M → N at a point c ∈ N is a homotopy invariant lower bound for the number of roots at c, that is, for the cardinality of f−1(c). There is a formula for calculating N(f ; c) if M and N are closed oriented manifolds of the same dimension. We extend the calculation of N(f; c) to manifolds that are not orientable, and also to manifolds that have non-empty boundaries and are not compact, provided that the map f is boundary-preserving and proper. Because of its connection with degree theory, we introduce the transverse Nielsen root number for maps transverse to c, obtain computational results for it in the same setting, and prove that the two Nielsen root numbers are sharp lower bounds in dimensions other than 2. We apply these extended root theory results to the degree theory for maps of not necessarily orientable manifolds introduced by Hopf in 1930. Thus we re-establish, in a new and modern treatment, the relationship of Hopf’s Absolutgrad and the geometric degree with homotopy invariants of Nielsen root theory, a relationship that is present in Hopf’s work but not in subsequent re-examinations of Hopf’s degree theory.

[1]  C GILYGIL,et al.  University of California, Los Angeles , 1963, Medical History.

[2]  Boju Jiang.  Fixed point classes from a differential viewpoint , 1981 .

[3]  R. Skora.  The degree of a map between surfaces , 1987 .

[4]  J. Jezierski,et al.  The Coincidence Nielsen Number on Non-Orientable Manifolds , 1993 .

[5]  J. Nielsen.  Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. II , 1927 .

[6]  H. Hopf.  Zur Topologie der Abbildungen von Mannigfaltigkeiten , 1930 .

[7]  R. Brooks,et al.  Nielsen numbers for roots of maps of aspherical manifolds. , 1995 .

[8]  T. Chiang.  The theory of fixed point classes , 1989 .

[9]  Kiang Tsai-han.  The theory of fixed point classes , 1989 .

[10]  RobertF Brown,et al.  A Lower Bound for the ?-Nielsen Number , 1969 .

[11]  William S. Massey,et al.  Algebraic Topology: An Introduction , 1977 .

[12]  G. E. Bredon.  Introduction to compact transformation groups , 1972 .

[13]  H. Kneser.  Glättung von Flächenabbildungen , 1928 .

[14]  Hellmuth Kneser,et al.  Die kleinste Bedeckungszahl innerhalb einer Klasse von Flächenabbildungen , 1930 .

[15]  H. Schirmer.  Nielsen theory of transversal fixed point sets (with an appendix: $C^∞$ and C0 fixed point sets are the same, by R. E. Greene) , 1992 .

[16]  H. Hopf.  Zur Topologie der Abbildungen von Mannigfaltigkeiten , 1928 .

[17]  A. Schwarz.  The Degree of a Map , 1994 .

[18]  RobertF Brown.  A middle-distance look at root theory , 1999 .

[19]  J. Nielsen,et al.  Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen , 1927 .

[20]  RobertF Brown,et al.  The Lefschetz fixed point theorem , 1971 .

[21]  L. Brouwer.  Über Abbildung von Mannigfaltigkeiten , 1911 .

[22]  L. Xiaosong.  On the root classes of mapping , 1986 .

[23]  Yongwu Rong,et al.  The preimages of submanifolds , 1992, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  D. Gonçalves,et al.  Equations in free groups and coincidence of mappings on surfaces , 2001 .

[25]  R. Brooks.  On the sharpness of the ∆2 and ∆1 Nielsen numbers. , 1973 .

[26]  P. Olum.  MAPPINGS OF MANIFOLDS AND THE NOTION OF DEGREE , 1953 .

[27]  Étale Groupoids,et al.  Homology Theory , 2009 .

[28]  J. Jezierski.  The Nielsen coincidence theory on topological manifolds , 1993 .

[29]  M. Postnikov.  Lectures in algebraic topology , 1983 .

[30]  RobertF Brown,et al.  Nielsen theory of roots of maps of pairs , 1999 .