Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStabilisation, bordism and embedded spheres in 4--manifolds
95
0
0.0
(
0
)
Authors
Christian Bohr
Ask ChatGPT about the research
No Arabic abstract
It is one of the most important facts in 4-dimensional topology that not every spherical homology class of a 4-manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4-manifold by adding products of 2-spheres, a process which is usually called stabilisation. In this paper, we extend this result to non-simply connected 4-manifolds and show how it is related to the Spin^c-bordism groups of Eilenberg-MacLane spaces.
rate research
Read More
A strategy for constructing an embedded sphere in a 4-manifold realizing a given homology class which has been successfully applied in the past is to represent the class as a first step stably by an embedded sphere, i.e. after adding products of 2-spheres, and to move that sphere back into the original manifold. In this paper, we study under what conditions the first step of this approach can be carried out if the 4-manifold at hand is not simply connected. One of our main results is that there are - apart from the well known Arf invariant - additional bordism theoretical obstructions to stably representing homology classes by embedded spheres.
We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples of pseudomanifold bordism theories, such as bordism of Witt spaces and IP spaces. Along the way, we relate the stratified and unstratified points of view for describing various (stratified) pseudomanifold classes.
We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. We recover the Rokhlin and Donaldsons Theorems from the Galois theory of the non-commutative fields.
In this article, we construct countably many mutually non-isotopic diffeomorphisms of some closed non simply-connected 4-manifolds that are homotopic to but not isotopic to the identity, by surgery along $Theta$-graphs. As corollaries of this, we obtain some new results on codimension 1 embeddings and pseudo-isotopies of 4-manifolds. In the proof of the non-triviality of the diffeomorphisms, we utilize a twisted analogue of Kontsevichs characteristic class for smooth bundles, which is obtained by extending a higher dimensional analogue of March{e}--Lescops equivariant triple intersection in configuration spaces of 3-manifolds to allow Lie algebraic local coefficient system.
In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4-manifold with even intersection form. Furthermore, we introduce a new geometric group invariant and discuss some of its properties.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
