Одељење за математику, 26. септембар 2014.

• 22. Септембар, 2014
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Састанак Одељења за математику биће одржан у петак, 26. септембра 2014. у сали 301ф са почетком у 14 часова.

Предавач: Ulrich Koschorke, University of Siegen

Наслов предавања: TOPOLOGICAL FIXED POINT AND COINCIDENCE THEORY

Садржај: In topological fixed point theory we are mainly interested in the following question. Can a given selfmap $ф$ of a manifold $M$ be deformed continuously until it has no fixed points? And we try to measure to what extent $f$ fails to be 'homotopically fixed point free' in  this sense. The classical Lefschetz number yields a necessary condition. But a much better measure is the Nielsen number of $f$. It vanishes precisely if $f$ is homotopic to a fixed point free map EXCEPT when $M$ is a surface with strictly negative Euler characteristic (in which case this statement can be dramatically wrong).

In coincidence theory we do not just compare a selfmap $f$ with the identity map but we study the coincidence set $C$ of an arbitrary pair $f,g$ of maps from the domain $M$ to a possibly different target manifold $N$ (i. e. $C$ is the set of points $x$ in $M$ where $f(x)=g(x)$). Can the maps $f,g$  be deformed away from one another? In other words: can $C$ be made empty by suitable homotopies?

Since the dimensions $m$ and $n$ of the domain and the target need not agree, generically the coincidence set $C$ will be an ($m$-$n$)-dimensional manifold (and not just consist of isolated points as in the fixed point setting). Thus the geometric methods of differential topology come into play, and deep notions such as bordism, Kervaire invariants, Hopf invariants etc. enter the picture. In particular, Nielsen numbers get a new, deeper meaning and answer some, but not all central questions (and may allow us to measure what can go wrong).

REFERENCE: U. Koschorke,  Minimum numbers and Wecken theorems in topological coincidence theory. I, J. Fixed Point Theory Appl. 10, 1 (2011), 3-36.