He studied at the École Polytechnique Fédérale in Zürich. He came under the influence of the topologist Heinz Hopf, and the Lie group theorist Stiefel. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of Leray and Henri Cartan.

He collaborated with Jacques Tits in fundamental work on algebraic groups, and with Harish-Chandra on their arithmetic subgroups. In an algebraic group G a *Borel subgroup* H is one such that the homogeneous space G/H is a projective variety, and as small as possible. For example if G is GL_{n} then we can take H to be the subgroup of upper triangular matrices. In this case it turns out that H is a maximal solvable subgroup, and that the *parabolic* subgroups P between H and G have a combinatorial structure (in this case the G/P are the various flag manifolds). Both those aspects generalize, and play a central role in the theory.

The Borel-Moore homology theory applies to general locally compact spaces, and is closely related to sheaf theory.

He published a number of books, including work on the history of Lie groups.