arXiv subjects

Maximilian Stegemeyer

Publications and source records attributed to Maximilian Stegemeyer.

11 recordsLinked to original sources

arXiv2026

Transport functions for principal bundles and Morse homology with differential graded coefficients

We study transport functions as a Morse-theoretical way of describing principal bundles. Transport functions are maps from the spaces of broken gradient flow lines to a topological group and they encode the transition functions of the principal bundle. We describe and extend a construction by Voigt that yields such transport functions and show that one can recover the principal bundle from the transport function. Using transport functions with values in a topological group $G$ and a differential graded module over the chains of $G$ we define a chain complex in the style of Barraud-Damian-Humilière-Oancea's Morse homology with differential graded coefficients. We prove that in many cases the homology of this complex is the homology of an associated bundle. In the case of smooth bundles transport functions arise also from parallel transport with respect to a connection and the corresponding DG Morse complex turns out to be isomorphic to a complex defined in the style of Barraud-Damian-Humilière-Oancea. We eventually consider certain aspects of the functoriality of our constructions.

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arXiv2025

Approaches to critical point theory via sequential and parametrized topological complexity

The Lusternik-Schnirelmann category of a space was introduced to obtain a lower bound on the number of critical points of a $C^1$-function on a given manifold. Related to Lusternik-Schnirelmann category and motivated by topological robotics, the topological complexity (TC) of a space is a numerical homotopy invariant whose topological properties are an active field of research. The notions of sequential and parametrized topological complexity extend the ideas of topological complexity. While the definition of TC is closely related to Lusternik-Schnirelmann category, the connections of sequential and parametrized TC to critical point theory have not been fully explored yet. In this article we apply methods from Lusternik-Schnirelmann theory to establish various lower bounds on numbers of critical points of functions in terms of sequential and parametrized TCs. We carry out several consequences and applications of these bounds, among them a computation of the parametrized TC of the unit tangent bundles of $(4m-1)$-spheres.

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arXiv2025

On the string topology of symmetric spaces of higher rank

The homology of the free and the based loop space of a compact globally symmetric space can be studied through explicit cycles. We use cycles constructed by Bott and Samelson and by Ziller to study the string topology coproduct and the Chas-Sullivan product on compact symmetric spaces. We show that the Chas-Sullivan product for compact symmetric spaces is highly non-trivial for any rank and we prove that there are many non-nilpotent classes whose powers correspond to the iteration of closed geodesics. Moreover, we show that the based string topology coproduct is trivial for compact symmetric spaces of higher rank and we study the implications of this result for the string topology coproduct on the free loop space.

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arXiv2025

String topology coproduct and Turaev cobracket on surfaces

The string topology coproduct on the homology of the free loop space of a closed manifold induces a string cobracket on $S^1$-equivariant homology. We give a complete computation of the string topology coproduct for surfaces of higher genus by describing an algorithm which computes the coproduct of a cyclic word in terms of generators of the fundamental group of the surface. We further show that the string cobracket is the negative of the Turaev cobracket.

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arXiv2025

Extensions of the loop product and coproduct, the space of antipodal paths and resonances of closed geodesics

We study the space of paths in a closed manifold $M$ with endpoints determined by an involution $f\colon M\to M$. If the involution is fixed point free and if $M$ is $2$-connected then this path space is the universal covering space of the component of non-contractible loops of the free loop space of $M/\mathbb{Z}_2$. On the homology of said path space we study string topology operations which extend the Chas-Sullivan loop product and the Goresky-Hingston loop coproduct, respectively. We study the case of antipodal involution on the sphere in detail and use Morse-Bott theoretic methods to give a complete computation of the extended loop product and the extended coproduct on even-dimensional spheres. These results are then applied to prove a resonance theorem for closed geodesics on real projective space.

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arXiv2025

String topology on the space of paths with endpoints in a submanifold

In this article we consider algebraic structures on the homology of the space of paths in a manifold with endpoints in a submanifold. The Pontryagin-Chas-Sullivan product on the homology of this space had already been investigated by Hingston and Oancea for a particular example. We consider this product as a special case of a more general construction where we consider pullbacks of the path space of a manifold under arbitrary maps. The product on the homology of this space as well as the module structure over the Chas-Sullivan ring are shown to be invariant under homotopies of the respective maps. This in particular implies that the Pontryagin-Chas-Sullivan product as well as the module structure on the space of paths with endpoints in a submanifold are isomorphic for two homotopic embeddings of the submanifold. Moreover, for null-homotopic embeddings of the submanifold this yields nice formulas which we can be used to compute the product and the module structure explicitly. We show that in the case of a null-homotopic embedding the homology of the space of paths with endpoints in a submanifold is even an algebra over the Chas-Sullivan ring.

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arXiv2024

Intersection Multiplicity in Loop Spaces and the String Topology Coproduct

The string topology coproduct is often perceived as a counterpart in string topology to the Chas-Sullivan product. However, in certain aspects the string topology coproduct is much harder to understand than the Chas-Sullivan product. In particular the coproduct is not homotopy-invariant and it seems much harder to compute. In this article we give an overview over the string topology coproduct and use the notion of intersection multiplicity of homology classes in loop spaces to show that the string topology coproduct and the based string topology coproduct are trivial for certain classes of manifolds. In particular we show that the string topology coproduct vanishes on product manifolds where both factors have vanishing Euler characteristic and we show that the based coproduct is trivial for total spaces of fiber bundles with sections. We also discuss implications of these results.

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arXiv2023

The string topology coproduct on complex and quaternionic projective space

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.

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arXiv2022

Geodesic complexity via fibered decompositions of cut loci

The geodesic complexity of a Riemannian manifold is a numerical isometry invariant that is determined by the structure of its cut loci. In this article we study decompositions of cut loci over whose components the tangent cut loci fiber in a convenient way. We establish a new upper bound for geodesic complexity in terms of such decompositions. As an application, we obtain estimates for the geodesic complexity of certain classes of homogeneous manifolds. In particular, we compute the geodesic complexity of complex and quaternionic projective spaces with their standard symmetric metrics.

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arXiv2022

Geodesic complexity of homogeneous Riemannian manifolds

We study the geodesic motion planning problem for complete Riemannian manifolds and investigate their geodesic complexity, an integer-valued isometry invariant introduced by D. Recio-Mitter. Using methods from Riemannian geometry, we establish new lower and upper bounds on geodesic complexity and compute its value for certain classes of examples with a focus on homogeneous Riemannian manifolds. Methodically, we study properties of stratifications of cut loci and use results on their structures for certain homogeneous manifolds obtained by T. Sakai and others.

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arXiv2021

On the String Topology Coproduct for Lie Groups

The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups.

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