The coincidence Lefschetz Theorem
Người báo cáo: Jean Paul BRASSELET (Institut de Mathématiques de Luminy)
Thời gian: 9h sáng thứ 5, 23/2
Địa điểm: Phòng 4 nhà A14, Viện Toán học
Tóm tắt: Given two maps $f$ and $g$ between compact oriented manifolds $M$ and $N$ of the same dimension, the coincidence points are defined as points $xin M$ such that $f(x)=g(x)$. At these points, one defines the coincidence index (intersection number of the graphs). Lefschetz coincidence Theorem says that the sum of indices iis equal to alternating sum of suitable matrix  traces. The Lefschetz fixed point formula is just the case $M=N$ and $g$ is the identity map $1_M$ on $M$. The result has been generalized in  the case of manifolds with different dimensions. M. Goresky and R. MacPherson have extended the Lefschetz fixed point theorem for singular varieties  in the context of intersection homology and with suitable hypothesis on the varieties and the maps. In this lecture, I will recall the main definitions and results concerning the Lefschetz coincidence indices and classes in the case of manifolds with same and possibly different dimensions. In the case of possibly singular varieties, I will recall the situation of the Goresky-MacPherson Lefschetz fixed point theorem for singular varieties. That leads to the Lefschetz coincidence Theorem in the case of singular varieties. I will give examples  to illustrate the results.


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