He studied at the ETH Zürich. He came under the influence of the topologist Heinz Hopf, and the Lie group theorist Eduard 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 Jean 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 GLn 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 homogenous spaces G/P are the various flag manifolds). Both those aspects generalize, and play a central role in the theory.
He published a number of books, including work on the history of Lie groups.
He died in Princeton. (He used to answer the question of whether he was related to Émile Borel alternately by saying he was a nephew, and no relation.)