Subject: Obituary for Roland Schwaenzl, 1952--2004 From: "Ross E. Staffeldt" Date: Fri, 15 Oct 2004 14:45:17 -0600 To: dmd1@lehigh.edu Roland Schw\"anzl died on July 29, 2004, at the age of 52. He had been seriously ill since April, 2004. Roland Schw\"anzl started his mathematical career at the University of Saarbr\"ucken, where he submitted his Master's thesis (Diplomarbeit) "The Burnside ring of SO(3)" in 1975. His mathematical studies were supervised by Tammo tom Dieck, first at Saarbr\"ucken and later at G\"ottingen. Roland Schwaenzl defended his Ph.D. thesis on July 31, 1979, at the University of Osnabrueck, Germany. For most of his career he held positions at the University of Osnabrueck, where he completed his "Habilitation" in mathematics in 1999. Roland's early work was in the area of equivariant topology. Then he made contributions to the theory of A-infinity and E-infinity ring spaces, motivated by questions raised by Waldhausen's development of the algebraic K-theory of topological spaces. Subsequently, he studied topological Hochschild homology and algebraic K-theory, motivated by the need to generalize foundational material to apply to K-theory of A-infinity rings. In the late 1980s Roland also became interested in development of the internet for mathematics, with his principal interests related to developing the internet as a library resource. From practical experience with the difficulties of realizing library-related projects, he led the development and accreditation of a program for a Master's degree in "Information Engineering", which is a joint offering of the University of Osnabrueck and the University of Twente in the Netherlands. At the time of his death he was actively involved in research in algebraic topology, in projects aimed at improving the usefulness of the internet, and in the advising of students in the Information Engineering program. Roland Schw\"anzl had many coauthors, and I am very happy to have been one of them. Typically, the excitement of work for him was the process of figuring out the answer to the problem. Although writing up results was less exciting for him, concern for the readership was evident when we discussed drafts of our papers. He read carefully and sought clarity, even when the work was unavoidably technical. Roland also a had a sense of humor. Travel adventures, of which he had many, were a favorite subject of relaxation breaks when we worked together. But you had to pay attention. Roland's sense of humor was often so dry, you would suddenly realize a string of jokes and humorous anecdotes had nearly passed you by. I am sure that he will be missed by many others. Themes of Roland's work in algebraic topology: As mentioned above, his early work was in the area of equivariant topology. The title of Roland's doctoral dissertation is ``Equivariant vector fields on manifolds", and Rainer Vogt and and Tammo tom Dieck are named as advisors. A related publication appeared in 1982 in Mathematische Zeitschrift. In 1975 tom Dieck had published his first paper extending the definition of the Burnside ring from finite groups to compact groups, and other early works of Roland deal with the Burnside ring of a compact Lie group. One of his later preprints dealing with equivariant algebraic topology, and the first work in algebraic K-theory, considered the Reidemeister torsion invariants introduced by Rothenberg to provide invariants for group actions that are not necessarily free. Roland's contribution provides a definition of Rothenberg's invariant that is more direct. As a byproduct of Roland's framing of the definition, one may extend Rothenberg's idea to define other invariants for which splitting and product formulas are valid. In the late 1970s Waldhausen introduced the algebraic K-theory of spaces and a host of questions about the appropriate formulation of the notions of A-infinity and E-infinity ring spaces came under scrutiny by K. Igusa, J.P. May, and M. Steinberger, and R. Steiner, among others. Working with Rainer Vogt, Roland Schw\"anzl made a number of contributions to these questions. In the paper "Homotopy invariance of A-infinity and E-infinity ring spaces", Schw\"anzl and Vogt adapted the universal algebra approach to homotopy invariant algebraic structures on topological spaces of Boardman and Vogt to answer some of these questions. One of their achievements was to formulate the definition of A-infinity ring space to avoid the explicit mention of distibutivity relations. This definition was not far removed from that of May, in the sense that any A-infinity space in the sense of May is also one in the sense of Schw\"anzl and Vogt, and that any Schw\"anzl-Vogt A-infinity space is homotopy equivalent to a May A-infinity ring space. The reduction in complication of the definition makes it possible to show eventually that a space homotopy equivalent to an A-infinity ring space is also an A-infinity ring space: the homotopy invariance of A-infinity ring spaces. A subsequent preprint "Matrices over homotopy ring spaces and algebraic K theory" that appeared in the Osnabrueck mathematical preprint series in 1984 showed that matrices over A-infinity ring spaces are also A-infinity ring spaces in a canonical way, constructed various equivalent versions of the algebraic K-theory of A-infinity and E-infinity ring spaces, proved Morita invariance, showed that the K-theory of an E-infinity ring space is another E-infinity ring space, and showed that a definition of the K-theory of a ring space proposed by May agreed with one proposed by Steiner. Algebraic K-theory became an additional theme of Roland's work in algebraic topology in the 1990s. Here the works generally aim to carry notions of the algebraic K theory of rings or exact categories over to algebraic K theory of A-infinity ring spaces. For example, he worked up with Fiedorowicz, Steiner, and Vogt a non-connective delooping of the algebraic K-theory of an A-infinity ring by a generalization of Wagoner's suspension construction. Other works with Fiedorowicz and Vogt applied this construction to to the Hermitian K-theory of A-infinity rings with involutions, extending earlier work of Burghelea and Fiedorowicz on rational Hermitian K-theory. For a third example, a paper with Gunnarson, Vogt, and Waldhausen developed an undelooped version of algebraic K theory, generalizing a construction of Gillet and Grayson. This construction is a tool for an analysis of Segal operations on the algebraic K-theory of a topological space by Gunnarson and Schw\"anzl. He also worked with me on a project to carry over to A-infinity rings results of Waldhausen on algebraic K-theory of generalized free products of ordinary rings. This project started with a paper on the approximation theorem and the K-theory of generalized free products, which rederived several of Waldhausen's results using techniques of the algebraic K-theory of categories with cofibrations and weak equivalences. Questions derived from K-theoretic considerations led to other papers on such subjects as closed simplicial and topological model category structures for the categories of A-infinity and E-infinity monoids and ring spaces and adjoining roots of unity to E-infinity ring spectra As algebraic K-theory supplied themes for Roland's work, so did the development of topological Hochschild homology. A paper with Fiedorowicz, Pirashvili, Vogt, and Waldhausen related MacLane homology and topological Hochschild homology in an explicit way. A paper with McClure and Vogt showed that the topological Hochschild homology spectrum of an E-infinity ing spectrum is the tensor product of the spectrum with the circle, the tensor product being calculated in the category of E-infinity ring spectra.