PUB - Publikationen an der Universität Bielefeld

Purely algebraic objects like abstract groups, coset spaces, and G-modules do not have a notion ofhole as do analytical and topological objects. However, equipping an algebraic object with a globalaction reveals holes in it and thanks to the homotopy theory of global actions, the holes can bedescribed and quantified much as they are in the homotopy theory of topological spaces. Part I ofthis article, due to the first author, starts by recalling the notion of a global action and describesin detail the global actions attached to the general linear, elementary, and Steinberg groups. Withthese examples in mind, we describe the elementary homotopy theory of arbitrary global actions,construct their homotopy groups, and revisit their covering theory. We then equip the setU mn(R)of all unimodular row vectors of lengthnover a ringRwith a global action. Its homotopy groupsπi(U mn(R)),i>0 are christened the vectorK-theory groupsKi+1(U mn(R)),i>0 ofU mn(R).It is known that the homotopy groupsπi(GLn(R))of the general linear group GLn(R)viewed as aglobal action are the VolodinK-theory groupsKi+1,n(R). The main result of Part I is an algebraicconstruction of the simply connected covering mapStUmn(R)→EUmn(R)whereEUmn(R)is thepath connected component of the vector(1,0,...,0)∈U mn(R). The result constructs the map as aspecific quotient of the simply connected covering mapStn(R)→En(R)of the elementary globalactionEn(R)by the Steinberg global actionStn(R). As expected,K2(U mn(R))is identified withKer(StUmn(R)→EUmn(R)). Part II of the paper provides an exact sequence relating stability forthe VolodinK-theory groupsK1,n(R)andK2,n(R)to vectorK-theory groups.