Even-dimensional l–monoids and L–theory

Sixt, Jörg¹

¹Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany

Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 479-515

Résumé

Surgery theory provides a method to classify n–dimensional manifolds up to diffeomorphism given their homotopy types and n ≥ 5. In Kreck’s modified version, it suffices to know the normal homotopy type of their n 2 –skeletons. While the obstructions in the original theory live in Wall’s L–groups, the modified obstructions are elements in certain monoids ln(Z[π]). Unlike the L–groups, the Kreck monoids are not well-understood.

We present three obstructions to help analyze θ ∈ l2k(Λ) for a ring Λ. Firstly, if θ ∈ l2k(Λ) is elementary (ie trivial), flip-isomorphisms must exist. In certain cases flip-isomorphisms are isometries of the linking forms of the manifolds one wishes to classify. Secondly, a further obstruction in the asymmetric Witt-group vanishes if θ is elementary. Alternatively, there is an obstruction in L2k(Λ) for certain flip-isomorphisms which is trivial if and only if θ is elementary.

Détail

DOI : 10.2140/agt.2007.7.479

Keywords: surgery theory, $L$-groups

Bibliographie

[1] P M Cohn, Algebra. Vol. 2, Wiley (1989)

[2] M H Freedman, F Quinn, Topology of 4-manifolds, Princeton Mathematical Series 39, Princeton University Press (1990)

[3] I Hambleton, M Kreck, On the classification of topological $4$-manifolds with finite fundamental group, Math. Ann. 280 (1988) 85

[4] I Hambleton, M Kreck, Cancellation, elliptic surfaces and the topology of certain four-manifolds, J. Reine Angew. Math. 444 (1993) 79

[5] I Hambleton, M Kreck, P Teichner, Nonorientable $4$-manifolds with fundamental group of order $2$, Trans. Amer. Math. Soc. 344 (1994) 649

[6] M Kreck, Bordism of diffeomorphisms and related topics\rm, With an appendix by N W Stoltzfus, Lecture Notes in Mathematics 1069, Springer (1984)

[7] M Kreck, Surgery and duality, Ann. of Math. $(2)$ 149 (1999) 707

[8] M Kreck, S Stolz, A diffeomorphism classification of $7$-dimensional homogeneous Einstein manifolds with $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$-symmetry, Ann. of Math. $(2)$ 127 (1988) 373

[9] M Kreck, S Stolz, Some nondiffeomorphic homeomorphic homogeneous $7$-manifolds with positive sectional curvature, J. Differential Geom. 33 (1991) 465

[10] F Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979) 55

[11] A Ranicki, The algebraic theory of surgery. I. Foundations, Proc. London Math. Soc. $(3)$ 40 (1980) 87

[12] A Ranicki, Exact sequences in the algebraic theory of surgery, Mathematical Notes 26, Princeton University Press (1981)

[13] A Ranicki, High-dimensional knot theory, Springer Monographs in Mathematics, Algebraic surgery in codimension 2, With an appendix by E Winkelnkemper, Springer (1998)

[14] A Ranicki, Algebraic and geometric surgery, Oxford Mathematical Monographs, Oxford Science Publications, Oxford University Press (2002)

[15] C T C Wall, Surgery on compact manifolds, Mathematical Surveys and Monographs 69, American Mathematical Society (1999)

[16] H E Winkelnkemper, Equators of manifolds and the action of $\Theta^n$, PhD thesis, Princeton University (1970)

[17] H E Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973) 45