SEMINAR ON WHITEHEAD TORSION

Bonn, Winter semester 2006-2007

Consider pairs (K, L) of finite CW complexes for which there exists a strong deformation retraction K → L. The Whitehead group Wh(L) of a finite CW complex L consists of the equivalence classes [K, L] of such pairs under the relation (K, L) ∼ (K′, L) if there is a formal deformation from K to K′ relative to L, with the sum defined by [K, L] + [K′, L] = [K ∪_{L} K′, L]. In fact the Whitehead group Wh(L) depends only on the fundamental group of the connected components of L, and can also be defined (and sometimes computed) by algebraic means. The Whitehead torsion τ(f) ∈ Wh(L) of a (cellular) homotopy equivalence f : L′ → L is zero if and only if f is a simple-homotopy equivalence, a notion weaker to f being a homeomorphism.

The aim of this seminar is to follow the book of Cohen [Co] to study simplehomotopy theory, the Whitehead group and the Whitehead torsion in details, ending with applications such as the s-cobordism Theorem and the classification of lens spaces. For practical information please visit the web-page for the seminar

(http://www.math.uni-bonn.de/people/ausoni/seminar-ws06-07.html).

List of talks

1. Introduction. Homotopy equivalence; Whitehead’s combinatorial approach to homotopy theory; CW-complexes. [Co, Chapter 1]

2. A geometric approach to homotopy theory, I. Formal deformations; mapping cylinders and deformations. [Co, Chapter 2, §4, §5]

3. A geometric approach to homotopy theory, II. The Whitehead group of a CW complex; simplifying a homotopically trivial CW pair. [Co, Chapter 2, §6, §7]

4. A geometric approach to homotopy theory, III. Matrices and formal deformations. [Co, Chapter 2, §8]

5. The Whitehead group, I. Algebraic conventions; the groups K_{0}R and K_{1}R of a ring; the Whitehead group Wh(G) of a group. [Mi, §1, Appendix 2] and [Co, §9, §10]

6. The Whitehead group, II. Some information about Whitehead groups; complexes with preferred bases. [Co, §11, §12] and [Mi, §6] 7. Milnor’s definition of torsion, I. Acyclic chain complexes; stable equivalence of acyclic chain complexes; definition of the torsion of an acyclic complex. [Co, §13, §14, §15]

8. Milnor’s definition of torsion, II. Milnor’s definition of torsion; characterization of the torsion of a chain complex; changing rings. [Co, §16, §17, §18] and [Mi, §3]

9. Whitehead torsion in the CW category, I. The torsion of a CW pair – definition; fundamental properties. [Co, §19, §20]

10. Whitehead torsion in the CW category, II. The natural equivalence of Wh(L) and ⊕_{j}Wh(π_{1}L_{j}); the torsion of a homotopy equivalence; product and sum theorems. [Co, §21, §22, §23]

11. Whitehead torsion in the CW category, III. The relationship between homotopy and simple-homotopy; invariance of torsion, h-cobordism and the Hauptvermutung. [Co, §24, §25] and [Hu, Part 2]

12. Lens spaces, I. Definition of lens spaces; the 3-dimensional spaces L_{p,q}; cell structure and homology groups; homotopy classification (1). [Co, §26, §27, §28, §29.1 – §29.3]

13. Lens spaces, II. Homotopy classification (2); simple-homotopy equivalence of lens spaces; the complete classification. [Co, §29.4 – §29.6, §30, §31].

14/15. Additional topics. Topological invariance of Whitehead torsion; the Wall finiteness obstruction. [Co, Appendix], [Ro, §1.7].

References

[Co] M. M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, vol. 10, Springer, 1973.

[Hu] J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, Benjamin, 1969.

[Mi] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.

[Ro] J. Rosenberg, Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147, Springer, 1996 (second edition).

Ch. Ausoni, Mathematisches Institut, Beringstr. 1, D-53115 Bonn.

   E-mail address: ausoni@math.uni-bonn.de
   URL: http://www.math.uni-bonn.de/people/ausoni/