arXiv subjects
Publications and source records attributed to Manuel Amann.
In C. R. Math. Acad. Sci. Paris 348 (2010) pp. 283--285 (arXiv:0811.0840) we constructed examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus. The reasoning was based on an equivariant surgery lemma which only holds under additional assumptions. To remedy the situation we give a construction using explicit equivariant surgeries.
We prove several long-standing conjectures by Chen--Nagano on cohomological descriptions of the cardinalities of maximal antipodal sets in symmetric spaces. We actually extend these conjectures to the setting of generalised symmetric spaces of finite abelian $p$-groups and verify them mostly in this broader context drawing upon techniques from equivariant cohomology theory.
It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly K\"ahler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly K\"ahler" naturally generalize compact K\"ahler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.
The toral rank conjecture speculates that the sum of the Betti numbers of a compact manifold admitting a free action of a torus of rank $r$ is bounded from below by $2^r$. Clearly, such an action yields a torus bundle, and, more generally, the same cohomological bound is conjectured for total spaces of suitable topological torus fibrations by F\'elix--Oprea--Tanr\'e. In this article we show that this generalised toral rank conjecture cannot hold by providing various different counter-examples to it (for each rank $r\geq 5$). In particular, we show that there are sequences of smooth nilpotent fibre bundles of nilmanifolds with fibre a torus of rank $r$ such that the quotient of the total dimensions of the cohomologies of total space and fibre even converges to $0$ with $r$ tending to infinity. We moreover prove that none of our torus fibrations can be realised by almost free torus actions. More precisely, in the depicted sequence the difference of the ranks of the torus fibres in the bundle and the toral ranks (which are constant one) even tends to infinity. Similar to recent examples by Walker (of a completely different nature) this shows that the toral rank conjecture is not likely to follow from "weaker conjectures or structures".
We provide various counter-examples to the long-standing so-called "Omnibus Conjecture" in Rational Homotopy Theory. That is, we show that a space with finite dimensional even-degree rational cohomology and finite dimensional spherical rational homology may indeed have infinite dimensional rational cohomology. Moreover, we also discuss "dual" versions and special cases of the conjecture.
We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing $k$-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are. This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro--Wilhelm our approach partly generalizes recent work by Berndt--Olmos on the index conjecture.
Motivated by prominent problems like the Hilali conjecture Yamaguchi--Yokura recently proposed certain estimates on the relations of the dimensions of rational homotopy and rational cohomology groups of fibre, base and total spaces in a fibration of rationally elliptic spaces. In this article we prove these estimates in the category of formal elliptic spaces and, in general, whenever the total space in addition has positive Euler characteristic or has the rational homotopy type of a homogeneous manifold (respectively of a known example) of positive sectional curvature. Additionally, we provide general estimates approximating the conjectured ones. Moreover, we suggest to study families of rationally elliptic spaces under certain asymptotics, and we discuss the conjectured estimates from this perspective for two-stage spaces.
Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that their isotropy action is equivariantly formal. In this article we show that $(\mathbb{Z}_2\oplus \mathbb{Z}_2)$-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.
We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group $G$ for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen--Macaulay. In fact, we present results in a slightly more general context. We extend the methods in this field by a conceptual approach on equivariant cohomology via rational homotopy theory using an explicit rational model for a double mapping cylinder.
We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.
An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive. In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) we suggest two new variations of equivariant formality: "MOD-formal actions" and "actions of formal core". We investigate and characterize these new terms in many different ways involving various tools from rational homotopy theory, Hirsch--Brown models, $A_\infty$-algebras, etc., and, in particular, we provide different applications ranging from actions on symplectic manifolds and rationally elliptic spaces to manifolds of non-negative sectional curvature. A major motivation for the new definitions was that an almost free action of a torus $T^n\curvearrowright X$ possessing any of the two new properties satisfies the toral rank conjecture, i.e. $dim H^*(X;Q)\geq 2^n$. This generalizes and proves the toral rank conjecture for actions with formal orbit spaces.
In this article we provide evidence for a well-known conjecture which states that quasi-isometric simply-connected nilpotent Lie groups are isomorphic. We do so by constructing new examples which are rigid in the sense that whenever they are quasi-isometric to any other simply-connected nilpotent Lie group, the groups are actually isomorphic.
This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds in dimensions up to 16. Many of the calculations are sharp and they require less symmetry than previous classifications. When restricted to certain classes of manifolds that admit non-negative curvature, these results imply diffeomorphism classifications. Also studied is a closely related family of manifolds called positively elliptic manifolds, and we prove the Halperin conjecture in this context for dimensions up to 16 or Euler characteristics up to 16.
We encode a compact Lie group action on a compact manifold by the Sullivan model of its Borel construction. We then prove that deciding whether this action is almost free is NP-hard.
In this short note we observe that the Hilali conjecture holds for two-stage spaces, i.e. we argue that the dimension of the rational cohomology is at least as large as the dimension of the rational homotopy groups for these spaces. We also prove the Hilali conjecture for a class of spaces which puts it into the context of fibrations.
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.
Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.
We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-dimensional simply-connected biquotients from a rational point of view. Moreover, we also investigate further manifolds of special holonomy, like manifolds of holonomy Spin(7) or Sp(n)\Sp(1) in special situations provided by rational ellipticity or geometric formality.