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Sebastian Goette

Publications and source records attributed to Sebastian Goette.

18 recordsLinked to original sources
arXiv2025

Path components of $\mathrm{G}_2$-moduli spaces may be non-aspherical

Starting from Joyce's generalised Kummer construction, we exhibit non-trivial families of $\mathrm{G}_2$-manifolds over the two dimensional sphere by resolving singularities with a twisted family of Eguchi-Hanson spaces. We establish that the comparison map $\mathcal{G}_2^{\mathrm{tf}}(M) /\!\!/ \mathrm{Diff}(M)_0 \rightarrow \mathcal{G}_2^{\mathrm{tf}}(M) / \mathrm{Diff}(M)_0$ is a fibration over each path components with Eilenberg Mac Lane spaces as fibres, which allows us to show that these families remain non-trivial in $\mathcal{G}_2^{\mathrm{tf}}(M) / \mathrm{Diff}(M)_0$. In addition, we construct a new invariant based on characteristic classes that allows us to show that different resolutions give rise to different elements in the moduli space.

math.GT↗
arXiv2020

Nu-invariants of extra-twisted connected sums

We analyse the possible ways of gluing twisted products of circles with asymptotically cylindrical Calabi-Yau manifolds to produce manifolds with holonomy G_2, thus generalising the twisted connected sum construction of Kovalev and Corti, Haskins, Nordstr\"om, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordstr\"om arXiv:1505.02734 in terms of fixpoint and gluing contributions, which include different types of (generalised) Dedekind sums. Surprisingly, the calculations involve some non-trivial number-theoretical arguments connected with special values of the Dedekind eta-function and the theory of complex multiplication. One consequence of our computations is that there exist compact G_2-manifolds that are not G_2-nullbordant.

math.GT↗
arXiv2020

Highly connected 7-manifolds, the linking form and non-negative curvature

In a recent article, the authors constructed a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature. Each member of this family is the total space of a Seifert fibration with generic fibre $S^3$ and, in particular, has the cohomology ring of an $S^3$-bundle over $S^4$. In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an $S^3$-bundle over $S^4$, the first time that any such spaces have been shown to admit non-negative sectional curvature.

math.DG↗
arXiv2018

Distinguishing G_2-manifolds

In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy G_2 metrics on a closed 7-manifold, or to distinguish G_2-manifolds that are homeomorphic but not diffeomorphic. We also describe the twisted connected sum and extra-twisted connected sum constructions used to realise G_2-manifolds for which the above invariants differ.

math.DG↗
arXiv2017

Highly connected 7-manifolds and non-negative curvature

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.

math.DG↗
arXiv2015

An analytic invariant of G_2 manifolds

We prove that the moduli space of holonomy G_2-metrics on a closed 7-manifold is in general disconnected by presenting a number of explicit examples. We detect different connected components of the G_2-moduli space by defining an integer-valued analytic refinement of the nu-invariant, a Z/48-valued defect invariant of G_2-structures on a closed 7-manifold introduced by the first and third authors. The refined invariant is defined using eta invariants and Mathai-Quillen currents on the 7-manifold and we compute it for twisted connected sums \`a la Kovalev, Corti-Haskins-Nordstr\"om-Pacini and extra-twisted connected sums as constructed by the second and third authors. In particular, we find examples of G_2-holonomy metrics in different components of the moduli space where the associated G_2-structures are homotopic and other examples where they are not.

math.GT↗
arXiv2013

Large time limit and local L^2-index theorems for families

We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2-index formulas. As applications, we prove a local L^2-index theorem for families of signature operators and an L^2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L^2-eta forms and L^2-torsion forms as transgression forms.

math.DG↗
arXiv2012

Exotic smooth structures on topological fibre bundles I

When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and stably, this is a complete invariant. We give a more or less complete and self-contained exposition of this theory which is a reformulation of some of the results of [7]. An important application is the computation of the Igusa-Klein higher Reidemeister torsion invariants of these exotic smooth structures. Namely, the higher torsion invariant is equal to the Poincar\'e dual of the image of the smooth structure class in the homology of the base. This is proved in the companion paper [11] written by the first two authors.

math.KT↗
arXiv2011

Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain 7-manifolds

We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application, we compute the Eells-Kuiper and t-invariants of certain cohomogeneity one manifolds that were studied by Dearricott, Grove, Verdiani, Wilking, and Ziller. In particular, we determine the diffeomorphism type of a new manifold of positive sectional curvature.

math.DG↗
arXiv2010

Kreck-Stolz invariants for quaternionic line bundles

We generalise the Kreck-Stolz invariants s_2 and s_3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spin-manifolds M of dimension 4k-1 with H^3(M; \Q) = 0 such that c_2(E)\in H^4(M) is torsion. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-spheres up to almost-diffeomorphism, that is, diffeomorphism up to connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds. The t-invariant also gives information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over HP^k. The t-invariant for S^{4k-1} is closely related to the Adams e-invariant on the (4k-5)-stem.

math.GT↗
arXiv2010

Exotic smooth structures on topological fibre bundles II

We use a variation of a classical construction of A. Hatcher to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension (at least twice the base dimension plus 3). Using a variation of the Dwyer-Weiss-Williams smoothing theory which we explain in a separate joint paper with Bruce Williams [11], we associate a homology class in the total space of the bundle to each exotic smooth structure and we show that the image of this class in the homology of the base is the Poincar\'e dual of the relative higher Igusa-Klein (IK) torsion invariant. This answers the question, in the relative case, of which cohomology classes can occur as relative higher torsion classes.

math.KT↗
arXiv2008

Torsion Invariants for Families

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.

math.DG↗
arXiv2007

Scalar Curvature Estimates by Parallel Alternating Torsion

We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups of equal rank, equipped with a normal homogeneous metric. We also correct a mistake in the treatment of odd-dimensional spaces in arXiv:math/0010199 and arXiv:0705.0500

math.DG↗
arXiv2006

Vafa-Witten Estimates for Compact Symmetric Spaces

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G-rk H\le 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.

math.DG↗
arXiv2003

Morse theory and higher torsion invariants II

Let p: M -> B be a family of compact manifolds equipped with a unitarily flat vector bundle F -> M. We generalize Igusa's higher Franz-Reidemeister torsion τ(M/B;F) to the case that the fibre-wise cohomology H^*(M/B;F) -> B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the difference of τ(M/B;F) and the higher analytic torsion \Cal T(M/B;F). We also generalise the examples given in math.DG/0111222 .

math.DG↗
arXiv2003

Morse theory and higher torsion invariants I

We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with Bismut. Under additional conditions on F, the torsion T is related to Igusa's higher Franz-Reidemeister torsion. As an application, we use the higher analytic torsion to detect infinite families of smooth bundles p_i: M -> B with diffeomorphic fibres that are homeomorphic but not diffeomorphic as bundles.

math.DG↗
arXiv2002

Diffeomorphism type of the Berger space

We compute the Eells-Kuiper invariant of the Berger manifold SO(5)/SO(3) and determine that it is diffeomorphic to the total space of an S^3-bundle over S^4. This answers a question raised by K. Grove and W. Ziller.

math.DG↗