The theory of vector bundles (and their characteristic classes) appears to have been standardized in the 20th century by all of the familiar names. Considering its substantial importance throughout geometry, I'm curious as to where/how the idea originated. I'm sure that they were "implicitly" present well before a proper definition was given, and I assume they were introduced as a generalization of the tangent bundle.

Also, considering the various perspectives on vector bundles in algebraic geometry, how were algebraic geometers pre-20th century thinking about vector bundles?

I appreciate any information! If possible, I'd love to see references to historic papers on the topic.


1 Answer 1


There are brief historical notes on the history of vector bundles and characteristic classes in section 6.5.4 of Luke-Mishchenko, Vector Bundles and Their Applications. A more detailed account is given by Zisman in the History of Topology volume, ch. 22, but it has to be filtered because he goes over history of general fibrations and fiber bundles, rather than vector bundles specifically. The same goes for chapter III of Dieudonne's History of Algebraic and Differential Topology. On the history of affine connections and gauge theory see When and how was the geometric understanding of gauge theories developed?, and for notation Notation for fiber bundles - why E for total space?

Algebraic geometers pre-20th century were not thinking about vector bundles or anything similar, that perspective emerged after Leray and Grothendieck. Even vector spaces only appeared at the end of 19-th century. Some anticipation can be vaguely identified in Darboux's moving trihedrons from Leçons sur la Théorie Générale des Surfaces (1889) and Elie Cartan's early work on Pfaffian systems (1899), see Where did Cartan introduce his notation for basis vectors and covectors? for references. Zisman names Cartan as the founding father of vector bundles specifically, and Seifert as initiating the general terminology:

"The other founder father is Elie Cartan. In a long series of papers published between 1922 and 1925, in relation to differential geometry and connections, the author engaged himself in computations which may seem a little repetitive today but where one can however see how E. Cartan was endeavouring to associate, in a more and more precise way a vector, affine or projective space with each point of a manifold, providing a coherent system of relations in order to link all these spaces together.... It is Herbert Seifert who created the term gefaserter Raum (fibre space) in his 1932 paper Topologie drei-dimensionale gefaserter Railme. That work, still motivated by the study of 3-dimensional manifolds, contains a definition of fibre spaces, sometimes called Seifert fibrations..."

According to Dieudonne, vector bundles first appeared explicitly in Whitney's work of 1935. Here is from Luke-Mishchenko on the history of characteristic classes:

"The study of bundles was probably started by Poincare with the study of nontrivial coverings. Fibrations appeared in connection with the study of smooth manifolds, then characteristic classes were defined and for a long time they were the main tool of investigation. The Stieffel–Whitney characteristic classes were introduced by Stieffel and Whitney in 1935 for tangent bundles of smooth manifolds. Whitney also considered arbitrary sphere bundles. These were $\!\!\!\mod 2$ characteristic classes. The integer characteristic classes were constructed by Pontryagin. He also proved the classification theorem for general bundles with the structure group $O(n)$ and $SO(n)$ using a universal bundle over a universal base such as a Grassmannian. For complex bundles, characteristic classes were constructed by Chern. This was the beginning of the general study of bundles and the period was well described in the book of Steenrod.

A crucial point of time came with the discovery by Leray of spectral sequences which were applied to the calculation of the homology groups of bundles. Another was at the discovery by Bott of the periodicity of the homotopy groups of the unitary and orthogonal groups. From this point, vector bundles occupied an important place in the theory of bundles as it was through them that nontrivial cohomology theories were constructed. It was through Bott periodicity that the Grothendieck constructions became so significant in their influence on the development of algebraic K–theory.


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