historical note on characteristic classes

Excerpt from the Preface to *Characteristic classes* by John Milnor and Jim Stasheff.

The theory of characteristic classes began in the year 1935 with almost simultaneous work by HASSLER WHITNEY in the United States and EDUARD STIEFEL in Switzerland. Stiefel’s thesis, written under the direction of Heinz Hopf, introduce and studied certain “characteristic” homology classes determined by the tangent bundle of a smooth manifold. Whitney, then at Harvard University, treated the case of an arbitrary sphere bundle. Somewhat later he invented the language of cohomology theory, hence the concept of a characteristic cohomology class, and proved the basic product theorem.

In 1942 LEV PONTRJAGIN of Moscow University began to study the homology of Grassmann manifolds, using a cell subdivision due to Charles Ehresmann. This enabled him to construct important new characteristic classes. (Pontrjagin’s many contributions to mathematics are the more remarkable in that he is totally blind, having lost his eyesight in an accident at the age of fourteen.)

In 1946 SHING-SHEN CHERN, recently arrived at the Institute for Advanced Study from Kunming in southwestern China, defined classes for complex vector bundles. In fact he showed that the complex Grassmann manifolds have a cohomology structure which is much easier to understand than that of the real Grassmann manifolds.

According to this site, E. Stiefel

was awarded his doctorate for his thesis Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten from ETH Zürich in 1935. He published the results of this thesis in a paper in 1936.

the 1936., is of course Stiefel’s paper on Stiefel-Whitney characteristic classes.

That sites quotes

- V Szebehely, D Saari, J Waldvogal, U Kirchgraber,
*Eduard L Stiefel (1909-1978)*, in Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics, Math. Forschungsinst., Oberwolfach, 1978, Celestial Mech.**21**(1) (1980), 3-4.

who say in the paper

… perhaps Stiefel’s most famous contribution to pure mathematics, was dedicated to a fundamental study of the theory of vector fields on manifolds. Generalising the classical notion of the Eulerian characteristic of a manifold, he introduced the idea of the characteristic classes.

In that paper,

- Eduard Stiefel,
*Richtungsfelder und Fernparallelismus in $n$-dimensionalen Mannigfaltigkeiten*, Comm. Math. Helv.**8**, 305-353 (1935-1936) perm URL

page 324, “Characteristik” (number) seems to appear for the first time:

Die Zahl a heisst due Characteristik des $m$-Feldes $\mathfrak{F}$ auf der Sphaere $S^r = S^{n-m}$.

- E. Stiefel at German wikipedia
- Eduard Stiefel,
*Über Richtungsfelder in den projektiven Räumen und einen Satz aus der reellen Algebra*, Comm. Math. Helv.*13*, 201-218 (1940-1941) perm URL

Pontrjagin’s 1942 paper is

- Lev Pontrjagin,
*Characteristic cycles on manifolds*, C. R. (Doklady) Acad. Sci. URSS (N.S.) 35, (1942). 34–37.

Revised on March 29, 2011 19:56:24
by Zoran Škoda
(192.68.254.6)