Algebraic K-Theory of Group Rings and Fields

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9803342Carlsson In this project the investigator will be studying conjectures ofNovikov and Borel, and proposing and working on a new conjectureconcerning the algebraic K-theory of fields. In the Novikov-Borelportion of the project, he will be attempting to replace geometricconditions (which have allowed the proof of these conjectures in anumber of cases) by more algebraic and homotopy theoretic conditionsconcerning the fundamental group. The methods here include thePedersen-Weibel``bounded K-theory,'' and a G-theoretic version thatstudies non-free modules over the group rings in question. If thisis sufficiently successful, it could lead to the solution of Borel'sconjecture that two closed manifolds whose universal cover is Euclideann-space are in fact homeomorphic. In the algebraic K-theory of fieldsportion, the investigator has developed a new version of theLichtenbaum-Quillen conjectures, which appears to have a chance ofidentifying the K-theory of fields ``on the nose'' (rather than insufficiently high dimensions) and which depends directly onthe representation theory of the Galois group rather than on the groupitself. One hopes that this will give new arithmetic applications ofK-theory. Work on this project will be directed toward two importantproblems. The first revolves around the study of manifolds, spaces(such as spheres, tori, etc.) in which every point has a neighborhoodthat resembles ordinary Euclidean space. One notion of equivalencefor such manifolds is homotopy equivalence, a relatively weakcondition. For instance, a circle and an annulus are homotopyequivalent, since one direction of the annulus can be compressed toobtain a circle. The other notion is homeomorphism, which is a verystrong condition, requiring that there be an explicit way to identifythe points of one manifold with those of the other so that each pointof one corresponds to exactly one in the other. The latter istypically very hard to verify, while homotopy equivalence isrelatively easy to verify. The conjecture of Borel asserts that for alarge class of manifolds (those whose universal cover is ordinaryEuclidean space) homotopy equivalence implies homeomorphism. Theinvestigator is approaching this conjecture via algebraic methods,involving algebraic invariants of the manifolds in question. Thesecond problem studies the so-called ``algebraic K-theory'' of fields(algebraic objects in which one has addition, multiplication, anddivision, like the rational or the real numbers). It proposes a newconjecture for a description of this K-theory, one which is strongerthan the earlier Lichtenbaum-Quillen conjectures. In this project, theinvestigator will be developing methods for verifying his conjecturein many cases. The hope is that studying this conjecture will givenew applications to arithmetic for this topological invariant.***