On the Contribution of Wu Wen-Tsün to Algebraic Topology

Abstract

The aim of this article is to present the contribution of Wu Wen-Tsun to Algebraic Topology and more precisely to the theory of characteristic classes. Several papers provide complete and welldocumented biography and academic career of Wu Wen-Tsun, in particular, Hudecek, 2014; O’Connor and Robertson, 2006; Wen-Tsun Wu’s Academic Career, 2006; Selected works of Wen-Tsun Wu, 2008. The author does not repeat the details provided in these papers concerning the Wu Wen-Tsun’s bibliography, we will just mention people involved in the Wu Wen-Tsun’s period in France. In addition to Wu Wen-Tsun’s papers, the Dieudonne’s book (Dieudonne, 1960) provides an excellent presentation of main results of Wu Wen-Tsun in Algebraic and Differential Topology. The author will use and abuse of this book (and refer to) when suitable. In the introduction, the author recalls mainly historical facts concerning the contribution of Wu Wen-Tsun to Algebraic Topology. The second section shows specifically the contribution of Wu Wen-Tsun to the Stiefel-Whitney classes and introduces the third section, dealing with the (real) Wu classes. The author provides definition, properties as well as further developments and generalizations of the Wu classes. The fourth and fifth sections are devoted to recent applications: In Cobordism theory and in Mathematical Physics. The author notices that Wu classes have been used as well in other domains, in particular surgery theory (Madsen and Milgram, 1979). The last section concerns the complex Wu classes and shows that the more recent Mather classes coincide with the previously defined complex Wu classes, that is a result from Zhou (1994) (see also Liu, 1996). This article is devoted to the contribution of Wu Wen-Tsun to the theory of Characteristic Classes, which coincides with his “French period” (1947–1951). However, speaking of Algebraic Topology, it is worthwhile to mention the important contribution of Wu Wen-Tsun to the Theory of realization of complexes or manifolds in Euclidean spaces and of embedding classes. That coincides with his return to China (1956–1965).