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 diffeomorphism or homeomorphism types of manifolds of dimension >= 5. In this paper, a modification of this theory is presented, where instead of fixing a homotopy type one considers a weaker information. Roughly speaking, one compares n-dimensional compact manifolds with topological spaces whose k-skeletons are fixed, 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 CP^n and one can use the modified theory to obtain information about their diffeomorphism type although the homotopy classification is not known. The theory reduces this classification result to the determination of complete intersections in a certain bordism group. The restrictions are: If d = d_1 ... d_r is the total degree of a complete intersection X^n_{d_1,..., d_r} of complex dimension n, then the assumption is, that for all primes p with p(p-1) <= n+1, the total degree d is divisible by p^{[(2n+1)/(2p-1)]+1}. Theorem A. Two complete intersections X^n_{d_1,.,d_r} and X^n_{d'_1,\ldots , d'_s} of complex dimension n>2 fulfilling the assumption above for the total degree are diffeomorphic if and only if the total degrees, the Pontrjagin classes and the Euler characteristics agree.Comment: 48 pages, published versio

h-cobordisms between 1-connected 4-manifolds

In this note we classify the diffeomorphism classes rel. boundary of smooth h-cobordisms between two fixed 1-connected 4-manifolds in terms of isometries between the intersection forms.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper1.abs.html Version 2: reference to previous work of T Lawson adde

On 55-manifolds with free fundamental group and simple boundary links in S5S^5

We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply this to the classification of simple boundary links of $3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed $5$-manifolds with free fundamental group and trivial second homology group.Comment: 20 page

Homotopy self-equivalences of 4-manifolds

We establish a braid of interlocking exact sequences containing the group of homotopy self-equivalences of a smooth or topological 4-manifold. The braid is computed for manifolds whose fundamental group is finite of odd order.Comment: Changes made to improve exposition following a referee's repor

On the Product in Negative Tate Cohomology for Finite Groups

Our aim in this paper is to give a geometric description of the cup product in negative degrees of Tate cohomology of a finite group with integral coefficients. By duality it corresponds to a product in the integral homology of $BG$: {$$H_{n}(BG,\mathbb{Z})\otimes H_{m}(BG,\mathbb{Z})\rightarrow H_{n+m+1}(BG,\mathbb{Z})$$} for $n,m>0$. We describe this product as join of cycles, which explains the shift in dimensions. Our motivation came from the product defined by Kreck using stratifold homology. We then prove that for finite groups the cup product in negative Tate cohomology and the Kreck product coincide. The Kreck product also applies to the case where $G$ is a compact Lie group (with an additional dimension shift).Comment: 13 page