Geometry & Topology

Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms

Ulrich Koschorke

Full-text: Open access

Abstract

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs (f1,f2) of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC(f1,f2) (and MC(f1,f2), resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f1,f2). Furthermore, we deduce finiteness conditions for MC(f1,f2). As an application we compute both minimum numbers explicitly in various concrete geometric sample situations.

The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f1,f2) into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf–Ganea homomorphisms which turn out to yield finiteness obstructions for MC.

Article information

Source
Geom. Topol., Volume 10, Number 2 (2006), 619-666.

Dates
Received: 29 September 2005
Revised: 9 March 2006
Accepted: 21 April 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799737

Digital Object Identifier
doi:10.2140/gt.2006.10.619

Mathematical Reviews number (MathSciNet)
MR2240900

Zentralblatt MATH identifier
1128.55002

Subjects
Primary: 55M20: Fixed points and coincidences [See also 54H25] 55Q25: Hopf invariants 55S35: Obstruction theory 57R90: Other types of cobordism [See also 55N22]
Secondary: 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 55P35: Loop spaces 55Q40: Homotopy groups of spheres

Keywords
coincidence manifold normal bordism path space Nielsen number Ganea-Hopf invariant

Citation

Koschorke, Ulrich. Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms. Geom. Topol. 10 (2006), no. 2, 619--666. doi:10.2140/gt.2006.10.619. https://projecteuclid.org/euclid.gt/1513799737


Export citation

References

  • M F Atiyah, R Bott, A Shapiro, Clifford modules, Topology 3 (1964) 3–38
  • S A Bogatyĭ, D L Gonçalves, Z H, Coincidence theory: the minimization problem, Tr. Mat. Inst. Steklova 225 (1999) 52–86 English translation: Proc. Steklov Inst. Math. 225 (1999) 45–77
  • R B S 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
  • R F Brown, Wecken properties for manifolds, from: “Nielsen theory and dynamical systems (South Hadley, MA, 1992)”, Contemp. Math. 152, Amer. Math. Soc., Providence, RI (1993) 9–21
  • R F Brown, H Schirmer, Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary, Topology Appl. 46 (1992) 65–79
  • O Cornea, New obstructions to the thickening of CW–complexes, Proc. Amer. Math. Soc. 132 (2004) 2769–2781
  • O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, Mathematical Surveys and Monographs 103, American Mathematical Society, Providence, RI (2003)
  • A Dold, D L Gonçalves, Self-coincidence of fibre maps, Osaka J. Math. 42 (2005) 291–307
  • L Fernández-Suárez, A Gómez-Tato, D Tanré, Hopf–Ganea invariants and weak LS category, Topology Appl. 115 (2001) 305–316
  • T Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965) 295–322
  • M J Greenberg, J R Harper, Algebraic topology, Mathematics Lecture Note Series 58, Benjamin/Cummings Publishing Co. Advanced Book Program, Reading, Mass. (1981)
  • A Hatcher, F Quinn, Bordism invariants of intersections of submanifolds, Trans. Amer. Math. Soc. 200 (1974) 327–344
  • P J Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955) 154–172
  • J Jezierski, The least number of coincidence points on surfaces, J. Austral. Math. Soc. Ser. A 58 (1995) 27–38
  • B Jiang, Fixed points and braids, Invent. Math. 75 (1984) 69–74
  • B J Jiang, Fixed points and braids. II, Math. Ann. 272 (1985) 249–256
  • U Koschorke, Vector fields and other vector bundle morphisms–-a singularity approach, Lecture Notes in Mathematics 847, Springer, Berlin (1981)
  • U Koschorke, Coincidence theory in arbitrary codimensions: the minimizing problem, Oberwolfach Reports (2004)
  • U Koschorke, Linking and coincidence invariants, Fund. Math. 184 (2004) 187–203
  • U Koschorke, Selfcoincidences in higher codimensions, J. Reine Angew. Math. 576 (2004) 1–10
  • U Koschorke, Geometric and homotopy theoretic methods in Nielsen coincidence theory, Fixed Point Theory and Appl. to appear (2006)
  • U Koschorke, Nielsen coincidence theory in arbitrary codimensions, J. Reine Angew. Math., to appear (2006)
  • U Koschorke, B Sanderson, Geometric interpretations of the generalized Hopf invariant, Math. Scand. 41 (1977) 199–217
  • G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer, New York (1978)
  • A Wyler, Sur certaines singularités d'applications de variétés topologiques, Comment. Math. Helv. 42 (1967) 28–48