A marriage of manifolds and algebra : the mathematical work of

It is no accident that Peter Landweber’s career closely matches the striking unification of algebra and topology provided by the theory of bordism of manifolds, especially complex bordism: his work has been at the heart of that interaction. Here I will briefly describe some of Landweber’s main contributions to this story. The starting point was Thom’s work [59], from 1954, in which he used transversality to identify bordism classes of closed smooth nmanifolds with what we today call the nth homotopy group of the Thom spectrum MO (though of course it was in part an attempt to express Thom’s arguments conveniently that later led to the concept of a spectrum) and then computed the homology of this (by the Thom isomorphism) and the homotopy (and actually the homotopy type). The result was that any mod 2 cohomology class was carried by a “singular manifold,” the image of the fundamental class of a smooth closed manifold under a map; and the bordism ring is a polynomial algebra over Z/2Z with one generator in each positive degree not one less than a power of 2. Thom also considered the oriented bordism ring Ω∗ = π∗(MSO). This was followed in 1960 by the use of the newly minted Adams spectral sequence [1], independently by Milnor [44] and by Novikov [51] to make the analogous computation of the complex bordism ring Ω ∗ = π∗(MU). The result was a polynomial algebra over Z with one