Matthias Kreck

Surgery and duality

Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension‚ 5. In this paper, a modiflcation of this theory is presented, where instead of flxing a homotopy type one considers a weaker information. Roughly speaking, one compares n-dimensional compact manifolds with topological spaces whose k-skeletons are flxed, where k is at least [n=2]. A particularly attractive example which illustrates the concept is given by complete intersections. By the Lefschetz hyperplane theorem, a complete intersection of complex dimension n has the same n-skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known. The theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group. This was under certain restrictions carried out in [Tr]. The restrictions are: If d = d1¢:::¢dr is the total degree of a complete intersection X n1;:::;dr of complex dimension n, then the assumption is, that for all primes p with p(pi1)• n+1, the total degree d is divisible by p [(2n+1)=(2pi1)]+1 . Theorem A. Two complete intersections X n

A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3) X SU(2) X U(1)-symmetry

On donne des exemples qui montrent que des espaces homogenes homeomorphes ne sont pas necessairement diffeomorphes

NONCONNECTED MODULI SPACES OF POSITIVE SECTIONAL CURVATURE METRICS

For a closed manifold M let 9\~(M) (resp. 9\~ic(M)) be the space of Riemannian metrics on M with positive sectional (resp. Ricci) cur- vature and let Diff(M) be the diffeomorphism group of M, which acts on these spaces. We construct examples of 7-dimensional manifolds for which the moduli space 9\~(M)/ Diff(M) is not connected and others for which 9\~c(M)/ Diff(M) has infinitely many connected components. The examples are obtained by analyzing a family of positive sectional curvature metrics on homogeneous spaces constructed by Aloff and Wallach, on which SU(3) acts transitively, respectively a family of positive Einstein metrics constructed by Wang and Ziller on homogeneous spaces, on which SU(3) x SU(2) x U(I) acts transitively. MAX-PLANCK-INSTITUT FUR MATHEMATIK, GOTTFRIED-CLAREN-STRASSE 26, 5300 BONN 3, GERMANY AND FACHBEREICH MATHEMATIK, UNIVERSITAT MAINZ, 6500 MAINZ, GERMANY Current address: Johannes Gutenberg Universitat Mainz, Fachbereich Mathematik, Staudinger- weg 9, 6500 Mainz, Germany E-mail address:kreck@topologie.mathematik.uni-mainz.de DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, NOTRE DAME, INDIANA 46556 E-mail address: stolz.l@nd.edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Nonorientable 4-manifolds with fundamental group of order 2

In this paper we classify nonorientable topological closed 4-manifolds with fundamental group Z/2 up to homeomorphism. Our results give a complete list of such manifolds, and show how they can be distinguished by explicit invariants including characteristic numbers and the ti-invariant associated to a normal PinC-structure by the spectral asymmetry of a certain Dirac operator. In contrast to the oriented case, there exist homotopy equivalent nonorientable topological 4-manifolds which are stably homeomorphic (after connected sum with S2 x S2) but not homeomorphic.

Counterexamples to the Kneser conjecture in dimension four

We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM0#M1 unlessM0 orM1 is homeomorphic toS4. LetN be the nucleus of the minimal elliptic Enrique surfaceV1(2, 2) and putM=N∪∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S2×S2) is diffeomorphic toM0#M1 for non-simply connected closed smooth four-manifoldsM0 andM1 if and only ifk≥8. On the other hand we show thatM is homeomorphic toM0#M1 for closed topological four-manifoldsM0 andM1 withπ1(Mi)=ℤ/2.

A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres

We give an elementary proof of the rational Hurewicz theorem and compute the rational cohomology groups of Eilenberg–MacLane spaces and the rational homotopy groups of spheres. Instead of using the Serre spectral sequence, we only assume the classical Hurewicz theorem, and give a short proof of the rational Gysin and Wang long exact sequences, which are applied inductively to the path fibration of Eilenberg–MacLane spaces.

On the cut-and-paste property of higher signatures of a closed oriented manifold

Abstract We extend the notion of the symmetric signature σ( M ,r)∈L n (R) for a compact n-dimensional manifold M without boundary, a reference map r : M→BG and a homomorphism of rings with involutions β : Z G→R to the case with boundary ∂M, where ( M , ∂M )→(M, ∂M) is the G-covering associated to r. We need the assumption that C ∗ ( ∂M ) ⊗ Z G R is R-chain homotopy equivalent to a R-chain complex D ∗ with trivial mth differential for n=2m resp. n=2m+1. We prove a glueing formula, homotopy invariance and additivity for this new notion. Let Z be a closed oriented manifold with reference map Z→BG. Let F⊂Z be a cutting codimension one submanifold F⊂Z and let F →F be the associated G-covering. Denote by α m ( F ) the mth Novikov–Shubin invariant and by b m (2) ( F ) the mth L2-Betti number. If for the discrete group G the Baum–Connes assembly map is rationally injective, then we use σ( M ,r) to prove the additivity (or cut and paste property) of the higher signatures of Z, if we have α m ( F )=∞ + in the case n=2m and, in the case n=2m+1, if we have α m ( F )=∞ + and b m (2) ( F )=0 . This additivity result had been proved (by a different method) in (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999, Corollary 0.4) when G is Gromov hyperbolic or virtually nilpotent. We give new examples, where these conditions are not satisfied and additivity fails. We explain at the end of the introduction why our paper is greatly motivated by and partially extends some of the work of Leichtnam et al. (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999), Lott (Math. Ann., 1999) and Weinberger (Contemporary Mathematics, 1999, p. 231).

Differential forms and 0-dimensional super symmetric field theories

We show that closed differential forms on a smooth manifold X can be interpreted astopological(respectivelyEudlidean)supersymmetricfieldtheoriesofdimension0j1overX. As a consequence, concordance classes of such field theories are shown to represent de Rham cohomology. The main contribution of this paper is to make all new mathematical notions regarding supersymmetric field theories precise.

h-cobordisms between 1-connected 4-manifolds

In this note we classify the dieomorphism classes rel. boundary of smooth h{ cobordisms between two xed 1{connected 4{manifolds in terms of isometries between the intersection forms.

Rohlin invariants, integrality of some spectral invariants and bordism of spin diffeomorphisms

Abstract According to Lee and Miller, the Rohlin invariant of a 0-bordant (8k+3)-dimensional Spin manifold is a spectral invariant. We prove that this invariant is integral even if the manifold is not 0-bordant. This integrality result is equivalent to the splitting of the exact sequence computing the bordism group of Spin diffeomorphisms. The Rohlin invariant of a Spin diffeomorphism is defined via its mapping torus. We give a product formula for the Rohlin invariant of Spin diffeomorphisms and a connection to the Oshanine invariant.