arXiv subjects

Markus Banagl

Publications and source records attributed to Markus Banagl.

At least 19 recordsLinked to original sources

arXiv2026

Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps. Part 2

In this Part 2 of our article we give a detailed discussion of the compatibility between the analytic Gysin maps we have defined in Part 1 and the topological Gysin maps defined by the second author. A significant role is played by a bordism-like description of K-homology due to Jakob which is closely related to the geometric K-homology theory of Baum and Douglas. We give a self-contained proof of the equivalence of the former with the analytic K-homology theory of Kasparov. As an intermediate step towards proving our main result we use Thom's transversality theorem to describe Gysin maps compatibly with Jakob's definition of K-homology.

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arXiv2026

Equivariant L-Classes of Atiyah-Singer-Zagier Type for Singular Spaces

If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds.

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arXiv2025

The G-signature Theorem on Witt spaces

Let G be a compact Lie group and let X be an oriented Witt G-pseudomanifold. Using intersection cohomology it is possible to define Sign(G,X) in R(G), the G-signature of X. Let g be an element in G. Assuming that the inclusion of the fixed point set associated to g is normally non-singular, we prove a formula for Sign(g,X), the G-signature of X computed at g, thus extending to Witt G-pseudomanifolds the fundamental result proved by Atiyah, Segal and Singer on smooth compact G-manifolds. Along the way, we give a detailed study of the fixed point set of a Thom-Mather G-space X and our main result in this direction is a sufficient condition ensuring that the fixed point set associated to G is included in X in a normally non-singular manner. This latter result provides many examples where our formula applies.

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arXiv2025

Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps

We introduce smooth atlas stratified spaces. We show that this class is closed under cartesian products; consequently, it is possible to define fiber bundles of smooth atlas stratified spaces. We describe the resolution of such a space to a manifold with fibered corners and use this result in order to prove that the class of smooth atlas stratified spaces coincides with that of Thom-Mather stratified spaces. We then consider Witt pseudomanifolds (such as singular complex algebraic varieties) where it is well-known that a bordism invariant signature is available and equal to the Fredholm index of a realization of the signature operator. To each oriented fiber bundle of stratified spaces, with Witt fibers, we assign a class in bivariant KK-theory (with 2 inverted). Kasparov multiplication by this element defines a Gysin map in analytic K-homology and one of our main results is that this Gysin map preserves the analytic signature class of Witt spaces. We prove in fact a more general result: functoriality for fiber bundles in the sense that if one fiber bundle is the composition of two others then the KK-class of the former is the Kasparov product of the classes of the latter. We also discuss this result for other Dirac-type operators satisfying an analytic Witt condition, for example the spin-Dirac operator on a fibration of psc-Witt spin pseudomanifolds. We next define the analytic Gysin map associated to an oriented normally non-singular inclusion of Witt spaces and prove that it also preserves the signature class. Finally, we relate the analytic signature class of a Witt space with the topological Siegel-Sullivan orientation. Specifically we show that if one applies the inverse of the second Adams operation to the Sullivan orientation and complexifies then one obtains our analytic signature class under the natural identification between analytic and topological K-homology.

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arXiv2024

The Equivariant L-Class of Pseudomanifolds

We construct an equivariant L-class for orientation preserving actions of a compact Lie group on a Whitney stratified compact oriented pseudomanifold that satisfies the Witt condition, for example on a compact pure-dimensional complex algebraic variety. The class lies in equivariant rational homology and its restriction to the trivial group is the Goresky-MacPherson L-class. For a smooth action on a manifold, the class is equivariantly Poincar\'e dual to the Hirzebruch L-class of the Borel homotopy quotient of the tangent bundle. We also provide a product formula under the equivariant K\"unneth isomorphism. If the group acts freely, then the equivariant L-class identifies with the Goresky-MacPherson L-class of the orbit space. The construction method rests on establishing Whitney (B)-regularity for finite-dimensional compact pseudomanifold approximations to the Borel construction, and on the author's Verdier-Riemann-Roch type formulae for the Goresky-MacPherson L-class.

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arXiv2024

Link Bundles and Intersection Spaces of Complex Toric Varieties

There exist several homology theories for singular spaces that satisfy generalized Poincar\'e duality, including Goresky-MacPherson's intersection homology, Cheeger's $L^2$ cohomology and the homology of intersection spaces. The intersection homology and $L^2$ cohomology of toric varieties is known. Here, we compute the rational homology of intersection spaces of complex 3-dimensional toric varieties and compare it to intersection homology. To achieve this, we analyze cell structures and topological stratifications of these varieties and determine compatible structures on their singularity links. In particular, we compute the homology of links in 3-dimensional toric varieties. We find it convenient to use the concept of a rational homology stratification. It turns out that the intersection space homology of a toric variety, contrary to its intersection homology, is not combinatorially invariant and thus retains more refined information on the defining fan.

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arXiv2023

Topological Gysin Coherence for Algebraic Characteristic Classes of Singular Spaces

Brasselet, the second author and Yokura introduced Hodge-theoretic Hirzebruch-type characteristic classes $IT_{1, \ast}$, and conjectured that they are equal to the Goresky-MacPherson $L$-classes for pure-dimensional compact complex algebraic varieties. In this paper, we show that the framework of Gysin coherent characteristic classes of singular complex algebraic varieties developed by the first and third author in previous work applies to the characteristic classes $IT_{1, \ast}$. In doing so, we prove the ambient version of the above conjecture for a certain class of subvarieties in a Grassmannian, including all Schubert subvarieties. Since the homology of Schubert subvarieties injects into the homology of the ambient Grassmannian, this implies the conjecture for all Schubert varieties in a Grassmannian. We also study other algebraic characteristic classes such as Chern classes and Todd classes (or their variants for the intersection cohomology sheaves) within the framework of Gysin coherent characteristic classes.

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arXiv2023

Transfer and the Spectrum-Level Siegel-Sullivan KO-Orientation for Singular Spaces

Integrally oriented normally nonsingular maps between singular spaces have associated transfer homomorphisms on KO-homology at odd primes. We prove that such transfers preserve Siegel-Sullivan orientations, defined when the singular spaces are compact pseudomanifolds satisfying the Witt condition, for example pure-dimensional compact complex algebraic varieties. This holds for bundle transfers associated to block bundles with manifold fibers as well as for Gysin restrictions associated to normally nonsingular inclusions. Our method is based on constructing a lift of the Siegel-Sullivan orientation to a morphism of highly structured ring spectra which factors through L-theory.

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arXiv2022

The Uniqueness Theorem for Gysin Coherent Characteristic Classes of Singular Spaces

We establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature terminating on genera of explicit characteristic subvarieties that we construct. It enables us e.g. to apply intersection theory of Schubert varieties to obtain a uniqueness result for such characteristic classes in the homology of an ambient Grassmannian. Our framework applies in particular to the Goresky-MacPherson L-class by virtue of the Gysin restriction formula obtained by the first author in previous work. We illustrate our approach for a systematic computation of the L-class in terms of normally nonsingular expansions in examples of singular Schubert varieties that do not satisfy Poincar\'{e} duality over the rationals.

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arXiv2022

Bundle Transfer of L-Homology Orientation Classes for Singular Spaces

We consider transfer maps on ordinary homology, bordism of singular spaces and homology with coefficients in Ranicki's symmetric L-spectrum, associated to block bundles with closed oriented PL manifold fiber and compact polyhedral base. We prove that if the base polyhedron is a Witt space, for example a pure-dimensional compact complex algebraic variety, then the symmetric L-homology orientation of the base, constructed by Laures, McClure and the author, transfers to the L-homology orientation of the total space. We deduce from this that the Cheeger-Goresky-MacPherson L-class of the base transfers to the product of the L-class of the total space with the cohomological L-class of the stable vertical normal microbundle.

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arXiv2020

Stratified Formal Deformations and Intersection Homology of Data Point Clouds

Intersection homology is a topological invariant which detects finer information in a space than ordinary homology. Using ideas from classical simple homotopy theory, we construct local combinatorial transformations on simplicial complexes under which intersection homology remains invariant. In particular, we obtain the notions of stratified formal deformations and stratified spines of a complex, leading to reductions of complexes prior to computation of intersection homology. We implemented the algorithmic execution of such transformations, as well as the calculation of intersection homology, and apply these algorithms to investigate the intersection homology of stratified spines in Vietoris-Rips type complexes associated to point sets sampled near given, possibly singular, spaces.

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arXiv2019

Gysin Restriction of Topological and Hodge-Theoretic Characteristic Classes for Singular Spaces

We establish formulae that explain how the topological Goresky-MacPherson characteristic L-classes as well as the Hodge-theoretic Hirzebruch characteristic classes defined by Brasselet, Schürmann and Yokura transform under Gysin restrictions associated to normally nonsingular embeddings of singular spaces. We find that both types of classes transform in the same manner. We give a first application of these formulae in obtaining algebraic rigidity results for topologically homeomorphic projective varieties.

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arXiv2019

Persistent Intersection Homology for the Analysis of Discrete Data

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional manifold. In such cases, persistent homology can be successfully employed to extract features, remove noise, and compare data sets. The underlying problems in some application domains, however, turn out to represent multiple manifolds with different dimensions. Algebraic topology typically analyzes such problems using intersection homology, an extension of homology that is capable of handling configurations with singularities. In this paper, we describe how the persistent variant of intersection homology can be used to assist data analysis in visualization. We point out potential pitfalls in approximating data sets with singularities and give strategies for resolving them.

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arXiv2018

The L-Homology Fundamental Class for IP-Spaces and the Stratified Novikov Conjecture

An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric $L$-spectrum of $\Z$, which is, up to weak equivalence, an $E_\infty$ ring map. Using this map, we construct a fundamental $L$-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces.

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arXiv2017

Topological and Hodge L-Classes of Singular Covering Spaces and Varieties with Trivial Canonical Class

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We apply this result in proving the Brasselet-Schürmann-Yokura conjecture for normal complex projective 3-folds with at most canonical singularities, trivial canonical class and positive irregularity. The conjecture asserts the equality of topological and Hodge L-class for compact complex algebraic rational homology manifolds.

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arXiv2016

Intersection Spaces, Equivariant Moore Approximation and the Signature

We generalize the first author's construction of intersection spaces to the case of stratified pseudomanifolds of stratification depth 1 with twisted link bundles, assuming that each link possesses an equivariant Moore approximation for a suitable choice of structure group. As a by-product, we find new characteristic classes for fiber bundles admitting such approximations. For trivial bundles and flat bundles whose base has finite fundamental group these classes vanish. For oriented closed pseudomanifolds, we prove that the reduced rational cohomology of the intersection spaces satisfies global Poincaré duality across complementary perversities if the characteristic classes vanish. The signature of the intersection spaces agrees with the Novikov signature of the top stratum. As an application, these methods yield new results about the Goresky-MacPherson intersection homology signature of pseudomanifolds. We discuss several nontrivial examples, such as the case of flat bundles and symplectic toric manifolds.

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arXiv2015

High-Dimensional Topological Field Theory, Positivity, and Exotic Smooth Spheres

In previous work, we proposed a general framework of positive topological field theories (TFTs) based on Eilenberg's notion of summation completeness for semirings. In the present paper, we apply this framework in constructing explicitly a concrete positive TFT defined on smooth manifolds of any dimension greater than 1. We prove that this positive TFT detects exotic smooth spheres. We show further that polynomial invariants (subject to boundary conditions) can be extracted from the state sum if the dimension of the cobordisms is at least 3.

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arXiv2015

Hodge Theory for Intersection Space Cohomology

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincaré duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well.

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