Algebraic K-Theory of Group Rings and Fields

小行星9803342 在这个项目中，研究者将研究诺维科夫和博雷尔的理论，并提出和研究一个关于代数K-场论的新理论。 在该项目的诺维科夫-博雷尔部分，他将试图取代几何条件（这使得证明这些假设在一些情况下）更多的代数和同伦理论条件有关的基本群体。 这里的方法包括Pedersen-Weibel的“有界K-理论”和研究群环上的非自由模的G-理论版本。 如果这是足够成功的，它可能导致解决博雷尔的猜想，两个封闭的流形，其普遍覆盖是欧几里得空间实际上同胚。 在场运动的代数K理论中，研究者发展了一种新的版本的Lichtenbaum-Quillen代数，它似乎有机会“在鼻子上”（而不是不够高的维度）识别场的K理论，并且它直接依赖于伽罗瓦群的表示理论，而不是群本身。 人们希望这将给K理论带来新的算术应用。 这个项目的工作将针对两个重要问题。 第一个围绕着流形、空间（如球体、环面等）的研究。其中每个点都有一个类似于普通欧几里得空间的邻域。 这种流形的一个等价概念是同伦等价，这是一个相对较弱的条件。 例如，一个圆和一个环是同伦等价的，因为环的一个方向可以压缩得到一个圆。 另一个概念是同胚，这是一个非常强的条件，要求有一个明确的方式来识别一个流形上的点与其他的点，使得一个流形上的每个点都恰好对应于另一个流形上的一个点。 后者通常很难验证，而同伦等价相对容易验证。 Borel的猜想断言，对于一个大类的流形（那些其泛覆盖是ordinaryEuclidean空间）同伦等价意味着同胚。 调查员是接近这个猜想通过代数方法，涉及代数不变量的流形的问题。 第二个问题研究所谓的“代数K理论”的领域（代数对象，其中一个有加法，乘法和除法，如合理或真实的数字）。 它提出了一个新的猜想来描述这个K-理论，这个猜想比以前的Lichtenbaum-Quillen猜想更强。 在这个项目中，调查员将开发方法来验证他在许多情况下的能力。 希望研究这个猜想能给这个拓扑不变量在算术上的新应用。

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.***