Volume 14 (2008)

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Minimizing the number of Nielsen preimage classes

Olga Frolkina

Geometry & Topology Monographs 14 (2008) 193–217

DOI: 10.2140/gtm.2008.14.193

arXiv: 0904.1411


We find conditions on topological spaces X, Y and nonempty subset B of Y which guarantee that for each continuous map f:X→ Y there exists a map g∼ f such that Nielsen preimage classes of g-1(B) are all topologically essential.

I feel very honoured to have the possibility to contribute a paper to this volume dedicated to the memory of an outstanding mathematician and a pleasant good-humoured person: Heiner Zieschang. In 2002–2003 in M V Lomonosov Moscow State University Heiner gave a series of lectures on fixed points and coincidence theory, which I was lucky to attend. In the same period I learned the German language at his seminars. During a nice voyage in summer 2003 from Moscow to Saint Petersburg, in which I was invited to take part, I made the acquaintance with his wife Ute and daughter Kim; two years later I met his other daughter Tanja. Heiner guided my study of coincidences, intersections and preimages during my visit in November–December 2004 in Ruhr-Universität Bochum. It was planned, to continue the project in 2005. But that hope was doomed to disappointment….


preimage problem, Nielsen preimage class, topological essentiality, Nielsen preimage number, minimum number of preimage classes

Mathematical Subject Classification

Primary: 54H99, 55M99

Secondary: 55S35


Received: 30 March 2006
Revised: 4 March 2007
Accepted: 18 April 2007
Published: 29 April 2008

Olga Frolkina
Chair of General Topology and Geometry
Faculty of Mechanics and Mathematics
M V Lomonosov Moscow State University
Leninskie Gori, 119991 Moscow, GSP-1