Algebraic homotopy theory, groups, and K-theory

Abstract

A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies Department of Mathematics. LetMk be the category of algebras over a unique factorization domain k, and let ind−Affk denote the category of pro-representable functions from Mk to the category E of sets. It is shown that ind−Affk is a closed model category in such a way that its associated homotopy category Ho(ind−Affk) is equivalent to the homotopy category Ho(S) which comes from the category S of simplicial sets. The equivalence is induced by functors Sk : ind−Affk → S and Rk : S→ ind−Affk. In an effort to determine what is measured by the homotopy groups πi(X) = πi(SkX) of X in ind−Affk in the case where k is an algebraically closed field, some homotopy groups of affine reduced algebraic groupsG over k are computed. It is shown that, if G is connected, then π0(G) = ∗ if and only if the group G(k) of k-rational points of G is generated by unipotents. A fibration theory is developed for homomorphisms of algebraic groups which are surjective on rational points which allows the computation of the homotopy groups of the universal covering groups of the simple algebraic subgroups of the associated semi-simple group G/R(G), where R(G) is the solvable radical of G. The homotopy groups of simple Chevalley groups over almost all fields k are studied. It is shown that the homotopy groups of the special linear groups Sln and of the symplectic groups Sp2m converge, respectively, to the K-theory and L-theory of the underlying field k. It is shown that there are isomorphisms π1(S1n) ∼= H2(Sln(k);Z) ∼= K2(k) for n ≥ 3 and almost all fields k, and π1(Sp2m) ∼= H2(Sp2m(k);Z) ∼= −1L(k) for m ≥ 1 and almost all fields k of characteristic 6= 2, where Z denotes the ring of integers. It is also shown that π1(Sp2m) ∼= H2(Sp2m(k);Z) ∼= K2(k) if k is algebraically closed of arbitrary characteristic. A spectral sequence for the homology of the classifying space of a simplicial group is used for all of these calculations. Thesis Supervisor: Dr. Roy R. Douglas