Algebraic K-Theory of Group Rings and Fields

在这个项目中，研究者将研究novikov和Borel的猜想，并提出和研究一个关于场的代数k理论的新猜想。在novikov - borell部分的项目中，他将尝试用更多关于基本群的代数和同伦理论条件来取代几何条件（在许多情况下，这些条件已经允许这些猜想的证明）。这里的方法包括pedersen - weibel的“有界k理论”，以及研究群环上的非自由模的g理论版本。如果这一方法足够成功，就可以解决Borel的猜想，即两个闭流形在欧几里得空间上实际上是同胚的。在场运动的代数k理论中，研究者开发了ichtenbaum- quillen猜想的一个新版本，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理论提供新的算术应用。这个项目的工作将针对两个重要的问题。第一个是围绕流形、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。这种流形等价的一个概念是同伦等价，一个相对弱的条件。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被压缩以得到一个圆。另一个概念是同胚性，这是一个非常强的条件，要求有一种明确的方法来识别一个流形的点与另一个流形的点，使得一个流形的每一个点都对应于另一个流形中的一个点。后者通常很难验证，而同伦等价则相对容易验证。Borel的猜想断言，对于一大类流形（其泛覆盖为普通欧氏空间的流形），同伦等价意味着同胚。研究者正在通过代数方法接近这个猜想，涉及所讨论的流形的代数不变量。第二个问题研究所谓的域的“代数k理论”（域是指具有加法、乘法和除法的代数对象，如有理数或实数）。它提出了一个新的猜想来描述这个k理论，一个比早期的Lichtenbaum-Quillen猜想更强的猜想。在这个项目中，研究者将在许多情况下开发验证他的猜想的方法。希望通过对这一猜想的研究，可以为该拓扑不变量的算法提供新的应用

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.***