Oberseminar Topologie (Sommersemester 2001 & Wintersemester 2001/2002)

Trace maps

The seminar on trace maps, which started in April 2001, ended after twenty-two lectures in January 2002.

Our goal is to understand a proof of a result of Bökstedt, Hsiang and Madsen about the assembly map in algebraic K-theory. The proof requires the construction of the so called topological cyclic homology and the cyclotomic trace, which is also of independent interest. More details on the subject are available below.

The seminar takes place every Monday from 15:00 (s.t.) to 17:00 (with the traditional tea/coffee break from 16:00 to 16:30) in SR5. [Schedule]

Based on positive experiences during the A¹-homotopy seminar, there is a weekly one-hour workshop, to discuss informally further topics and background information related to the subject. It takes place Tuesday from 11:00 (s.t.) to 12:00 in SFB (room 205).

If you have any further question or if you want to join the mailing list for the seminar please do not hesitate to contact us.

Holger Reich	reichh@math.uni-muenster.de
Marco Varisco	varisco@uni-muenster.de

More information on the subject

The seminar is concerned with trace maps out of algebraic K-theory. The ultimate goal of the seminar is to understand a proof of the following result of Bökstedt, Hsiang and Madsen which discovers a big portion inside $K_n(\mathbb{Z}\Gamma)$ .

Theorem ([BHM]) Let $\Gamma$ be a group whose homology in each degree is a finitely generated abelian group. Then the rationalized assembly map in algebraic K-theory

$H_n(B\Gamma,\mathbb{K}(\mathbb{Z}))\otimes_{\mathbb{Z}}\mathbb{Q} \to K_n(\mathbb{Z}\Gamma)\otimes_{\mathbb{Z}}\mathbb{Q}$

is injective.

Here the left hand side admits a more explicit description in terms of group homology with rational coefficients. The result fits into the general philosophy of assembly maps. Among similar statements it has an astonishingly weak assumption on the group.

Its proof uses the cyclotomic trace map from K-theory to the so called topological cyclic homology TC, which has proven to be a powerful tool also in other K-theory computations. Topological cyclic homology will hence be the main object of study in our seminar. It is at least philosophically related to Connes' cyclic homology. But its construction works not on the level of homological algebra but on the level of spaces (or spectra). The definition involves some unexpected structure which is not visible anymore after passage to homological algebra.

Recently the definition of TC and the cyclotomic trace has been extended to schemes. Therefore the seminar could be appealing to algebraic geometers with an interest in algebraic K-theory.

To get a first impression one should read the introduction of Ib Madsen's survey article [M]. Moreover we highly recommend the notes [R1] of John Rognes, where the construction of topological cyclic homology and the cyclotomic trace is sketched on 14 pages without hiding any technical difficulties.

References

A file containing the following articles can be found in the library on the 5th floor.

[R1]: John Rognes.
Notes on topological cyclic homology and the cyclotomic trace map.
1996, http://www.math.uio.no/~rognes/papers/tcnotes.dvi.

[R2]: John Rognes.
On trace maps.
1998, http://www.math.uio.no/~rognes/papers/trace_maps.dvi.

[R3]: John Rognes.
Topological cyclic homology of S-algebras.
1998, http://www.math.uio.no/~rognes/papers/tc.dvi.

[M]: Ib Madsen.
Algebraic K-theory and traces.
In Current developments in mathematics, 1995 (Cambridge, MA), pages 191--321. Internat. Press, Cambridge, MA, 1994. [MR 98g:19004]

[BHM]: M. Bökstedt, W. C. Hsiang, and I. Madsen.
The cyclotomic trace and algebraic K-theory of spaces.
Invent. Math., 111(3):465--539, 1993. [MR 94g:55011]

[G]: Thomas G. Goodwillie.
The differential calculus of homotopy functors.
In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 621--630, Tokyo, 1991. Math. Soc. Japan. [MR 93g:55015]

[BCCGHM]: M. Bökstedt, G. Carlsson, R. Cohen, T. Goodwillie, W. C. Hsiang, and I. Madsen.
On the algebraic K-theory of simply connected spaces.
Duke Math. J., 84(3):541--563, 1996. [MR 97h:19002]

[J]: John D. S. Jones.
Cyclic homology and equivariant homology.
Invent. Math., 87(2):403--423, 1987. [MR 88f:18016]

[HM]: Lars Hesselholt and Ib Madsen.
On the K-theory of finite algebras over Witt vectors of perfect fields.
Topology, 36(1):29--101, 1997. [MR 97i:19002]

[DGMC]: B. I. Dundas, T. Goodwillie, and R. McCarthy.
The local structure of algebraic K-theory.
http://www.math.ntnu.no/~dundas/indexeng.html.

[B1]: Marcel Bökstedt.
Topological Hochschild homology.
Preprint, Bielefeld, 1988.

[B2]: Marcel Bökstedt.
Topological Hochschild homology of Z and Z/p.
Preprint, Bielefeld, 1988.

Schedule

April 23, 2001	Hochschild homology, spectra and topological Hochschild homology - I	Arthur Bartels
April 30, 2001	Hochschild homology, spectra and topological Hochschild homology - II	Juliane Sauer
May 7, 2001	Hochschild homology, spectra and topological Hochschild homology - III	Michel Matthey
May 14, 2001	Cyclic sets, cyclic homology and topological cyclic homology - I	Holger Reich
School and conference on high-dimensional manifold topology - ICTP, Trieste (Italy)
June 11, 2001	Cyclic sets, cyclic homology and topological cyclic homology - II	Thomas Schick
June 18, 2001	Cyclic sets, cyclic homology and topological cyclic homology - III	Roman Sauer
June 25, 2001	Equivariant spectra and computations	Michael Joachim
July 2, 2001	Algebraic K-theory	Jens Hornbostel
July 9, 2001	Trace maps	Marco Varisco
July 16, 2001	The algebraic K-theory Novikov conjecture - a strategy for the proof	Wolfgang Lück

[SFB] July 10, 2001	The cyclotomic trace and curves in K-theory	Christian Schlichtkrull (Stanford)
Summer break
October 29, 2001	Localization and completion of (modules, spaces and) spectra - I	Arthur Bartels
November 5, 2001	Localization and completion of (modules, spaces and) spectra - II	Michael Joachim
November 12, 2001	Equivariant spectra, transfer, and tom Dieck-Segal splitting - I	Wolfgang Lück
November 19, 2001	Equivariant spectra, transfer, and tom Dieck-Segal splitting - II	Markus Szymik (Bielefeld)
November 26, 2001	Rückblick und Vorschau	Marco Varisco & Holger Reich
December 3, 2001	How to compute the topological cyclic homology of a space	Roman Sauer
December 10, 2001	Dundas-McCarthy's approach to trace maps, and survey on relative theorems - I	Holger Reich
December 17, 2001	Dundas-McCarthy's approach to trace maps, and survey on relative theorems - II	Marco Varisco
January 7, 2002	Soulé's theorem - I	Holger Reich
January 14, 2002	Soulé's theorem - II	Matthias Strauch
January 21, 2002	Soulé's theorem - III	Holger Reich

Marco Varisco - January 22, 2002.
http://www.math.uni-muenster.de/u/varisco/seminar