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Christian Bohr

Publications and source records attributed to Christian Bohr.

9 recordsLinked to original sources
arXiv2002

Stabilisation, bordism and embedded spheres in 4--manifolds

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.

math.GT↗
arXiv2001

Embedded spheres and 4-manifolds with spin coverings

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.

math.GT↗
arXiv2001

On the kinkiness of closed braids

In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel knots cannot be unknotted using only positive crossing changes. We also describe a subgroup of infinite rank in the smooth knot concordance group of which no element has a strongly quasipositive representative.

math.GT↗
arXiv2001

Homology cobordism and classical knot invariants

In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the Z/2-homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given Z/2-homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot.

math.GT↗
arXiv2000

On the signatures of even 4-manifolds

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.

math.GT↗
arXiv2000

Immersions of surfaces in almost complex 4-manifolds

In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provide a generalization of a classical genus estimate due to V.A. Rokhlin to the case of immersed surfaces.

math.GT↗
arXiv1998

On the relation between lifting obstructions and ordinary obstructions

We consider partial liftings of maps at fibrations and compare the primary obstruction to extend the lifting with the obstruction to extend the lifting as a simple map into the total space. A relation between these two obstructions is proved for the case when the fiber is an Eilenberg-MacLane space. Furthermore it is shown that this result specialises to well known facts about secondary obstructions.

math.AT↗
arXiv1998

Embedded surfaces and almost complex structures

We prove necessary and sufficient conditions for a smooth surface in a 4-manifold X to be pseudoholomorphic with respect to some almost complex structure on X. This provides a systematic approach to the construction of pseudoholomorphic curves that do not minimize the genus in their homology class.

math.DG↗