# Research Area A3

# Facets of geometry: manifolds, dynamics, and curvature

## PIs: Hamenstädt, Huybrechts, Klemm, Lück, Sturm, Teichner

## Contributions by Ray

## Topics and goals

This research area aims for deep insights about manifolds, their automorphism groups and moduli spaces, and generalizations to metric measure spaces. This requires tools from various fields such as surgery theory, algebraic K-theory, measured and geometric group theory, harmonic analysis, and L2-invariants. We will combine forces to transfer techniques from one area to another. Among our main goals are the extension of the Farrell–Jones conjecture to reductive p-adic groups and new classification results for four-dimensional manifolds.

## State of the art, our expertise

**The Farrell–Jones conjecture**. This conjecture has been proved by Lück and his coauthors for the algebraic K- and L-theory of group rings for a large class of discrete groups, for instance hyperbolic groups, CAT(0)-groups, lattices in locally compact second countable Hausdorff groups, and fundamental groups of three-manifolds. They have given striking applications to the classification of manifolds and their automorphism groups as well as to geometric group theory, e.g., to hyperbolic groups with spheres as boundary and to fibering manifolds [BFL14, BL12, BLRR14].

**L ^{2}-invariants**. The Lück approximation theorem relates L2-Betti numbers to ordinary Q-Betti numbers. Partial results in characteristic p have been obtained by Lück and his coauthors. A systematic study of twisting L2-invariants by not necessarily unitary finite-dimensional representations has been initiated in [Lüc17]. Joint work with Friedl has led in dimension three to interesting connections between L2-torsion twisted by a one-parameter family of such representations and the Thurston polytope using the proof of Thurston’s virtual fibering conjecture [FL17].

**Low-dimensional manifolds**. The failure of the s-cobordism theorem and surgery theory in dimensions < 5 implies that techniques such as the ones arising from the work on the Farrell–Jones conjecture have to be supplemented in low dimensions. For example, the usual intersection form is enriched by the theory of Whitney towers which was developed by Teichner and his coauthors over the last years. Whitney towers are understood inductively by higher-order intersection invariants _ , see the figure, with values in a certain group generated by trivalent trees [CST14]. In the four-ball, Whitney towers with boundary give geometric filtrations of classical links. It is proven in [CST12] that these filtrations can be computed by Milnor invariants, Sato–Levine invariants and higher-order Arf invariants. The theory has been applied to the classification of string links and gives a geometric understanding of Cochran’s link invariants [CST17]. Schneiderman–Teichner [ST14] showed that for an arbitrary four-manifold M the vanishing of the non-repeating part of the higher-order intersection invariants is equivalent to representing homotopy classes in _2(M) by disjoint two-spheres.

**Moduli spaces and mapping spaces**. The classical moduli space of curves and of abelian differentials has been studied with tools from algebraic and differential geometry and from dynamical systems. Geometric group theory and low-dimensional topology are linked; for example, mapping class groups and three-manifolds are related by the mapping torus of a surface automorphism. Hamenstädt’s contribution to this very fast developing area mainly concentrates on the dynamical aspects of the theory [Ham13] and on the link between metrics on Teichmüller spaces and the cohomology of mapping class groups [Ham14]. Dynamical as well as arithmetic aspects also play a central role in Huybrechts’ study of moduli spaces of hyperkähler manifolds and K3 surfaces [Huy12]. Related from the perspective of topological string theory is Klemm’s work on topological A- and B-models on Calabi–Yau manifolds [KKP16].

The study of harmonic maps with values in metric spaces of nonpositive curvature, so-called CAT(0)-spaces, was initiated by Korevaar–Schoen and Jost; recently it was enhanced by Zhang– Zhu. The most natural setting for domain spaces are CD spaces, i.e., metric measure spaces with synthetic lower bounds on the Ricci curvature. Much progress about harmonic functions and heat flows on such spaces was made in the last decade, e.g., for Finslerian spaces by Ohta– Sturm [OS14].

## Research program

**The Farrell–Jones conjecture**. We expect many new applications of the Farrell–Jones conjecture to geometric and topological questions, for instance to automorphism groups of manifolds and to the classification of certain classes of manifolds such as torus bundles over lens spaces. We want to identify a group, or a property of groups, which has the potential to give a counterexample. A new long-term project is to formulate the algebraic K-theoretic analogue for a reductive p-adic group G. This would be the first instance where the Farrell–Jones conjecture is considered for a topological group. A potential proof will exploit the CAT(0)-structure and the resulting flow space on the associated Bruhat–Tits building. A consequence will be that the canonical map from colimK_G K0(H(K)) to K0(H(G)) is a bijection, where K runs through the compact open subgroups of G and H denotes the global Hecke algebra. This is closely related to the theory of smooth representations of reductive p-adic groups, as such representations correspond to modules over the global Hecke algebra, and thus links to RA A2. Another topic is to formulate a version of the Baum–Connes conjecture for Fréchet group algebras and prove it for classes of groups, e.g., SLn(Z), for which the Baum–Connes conjecture is not known. Computations of the K- and L-groups for specific groups, which will have consequences to questions in geometry, topology and operator algebras, will be carried out using methods from equivariant homotopy theory; see also RA A2.

**L2-invariants**. A concrete goal concerning L2-invariants is to attack the Lück approximation theorem in characteristic p and the conjecture that the first L2-Betti number of a group is equal to its cost and to the rank gradient. The Bergeron–Venkatesh conjecture relates the L2-torsion to the growth of the torsion of the homology. Lück [Lüc16] proposed both algebraic and analytic methods for its proof. The conjecture is especially interesting in dimension three, where it relates the torsion growth of the first homology to the volume. Hamenstädt intends to use methods from low-dimensional topology by investigating manifolds fibering over the circle and to develop an effective Cˇ ech cohomology model on graphs for the thick part of a fibered hyperbolic threemanifold. Twisted L2-torsion shall be investigated for three-manifolds and knots as a function on the representation space of finite-dimensional linear representations of their fundamental groups. Its impact on high-dimensional manifolds and geometric groups will be investigated.

**Low-dimensional manifolds**. Ray and Teichner will study knot and link concordance and its relation to the classification of four-manifolds. The focus will be on fundamental groups for which the four-dimensional topological s-cobordism theorem holds, on smooth versus topological disks in the four-ball and on finding counterexamples to the smooth four-dimensional Poincaré conjecture. Using recent progress in a Bousfield–Kan spectral sequence that describes configuration space approximations to links, Teichner wants to understand the possible values for his new higher-order Arf invariants and solve the remaining problems concerning Whitney towers in the four-ball. He will try to generalize the spectral sequence to concordance spaces to find new obstructions for representing classes in _2(M4) by embeddings of two-spheres. This is intimately related to the classification of smooth four-manifolds, the main open problem in manifold topology. A joint goal of Lück and Teichner is a general classification in the stable case, where one allows connected sums with S2 _ S2. Teichner conjectures that in the spin case, algebraic invariants like the (stable, equivariant) intersection form and a cubic form lead to a complete stable classification. This has been achieved for abelian and two-dimensional fundamental groups by Teichner and coauthors, the four-torus being a particularly interesting case. For other aspherical four-manifolds, the Farrell–Jones conjecture will be an important tool in this classification.

**Moduli spaces and mapping spaces**. The differential geometry of moduli space of curves will be further developed by Hamenstädt. A main goal is a better geometric understanding of the Schottky locus via its intersection with specific Hilbert modular varieties determined by pseudo- Anosov maps with prescribed trace field. The intriguing similarities between the moduli space of curves and moduli spaces of K3 surfaces, with relations to RA A1 and IRU D2, and of hyperkähler manifolds will be a focus of Huybrechts’ work. He also aims for new topological boundedness and finiteness results for hyperkähler manifolds. Many of these aspects also permeate Klemm’s further attempts to calculate topological string amplitudes.

The study of mappings from CD-spaces into CAT(0)-spaces will be pushed forward by Sturm. Existence and regularity results, especially Lipschitz continuity, of energy minimizers shall be derived as well as gradient estimates and equilibration rates for the associated time evolution. Moreover, in collaboration with RA B1 also suitable random evolutions on mapping spaces, in particular on loop spaces, will be studied.

## Summary

RA A3 will advance the understanding of manifolds, their automorphism groups and moduli spaces, and of metric measure spaces. One important innovative aspect is to combine the PIs’ expertise and to transfer tools between separate fields such as surgery theory, algebraic K- theory, measured and geometric group theory, harmonic analysis, and L2-invariants. Examples are the desired proofs of the Farrell–Jones conjecture using input from CAT(0)-spaces, controlled topology, homotopy theory, and dynamical systems, and proofs of new classification results in low-dimensional topology using algebraic and geometric methods as well as L2-invariants.

## Bibliography

[BFL14] A. Bartels, F. T. Farrell, and W. Lück. The Farrell–Jones Conjecture for cocompact lattices in virtually connected Lie groups. *J. Amer. Math. Soc., 27(2):339–388, 2014*.

[BL12] A. Bartels and W. Lück. The Borel conjecture for hyperbolic and CAT(0)-groups.* Ann. of Math. (2), 175:631–689, 2012*.

[BLRR14] A. Bartels, W. Lück, H. Reich, and H. Rüping. K- and L-theory of group rings over GLn(Z). *Inst. Hautes Études Sci. Publ. Math., 119:97–125, 2014*.

[CST12] J. Conant, R. Schneiderman, and P. Teichner. Whitney tower concordance of classical links. Geom. *Topol., 16(3):1419–1479, 2012*.

[CST14] J. Conant, R. Schneiderman, and P. Teichner. Milnor invariants and twisted Whitney towers. J. Topol., *7(1):187–224, 2014*.

[CST17] J. Conant, R. Schneiderman, and P. Teichner. Cochran’s i-invariants via twisted Whitney towers. *J. Knot Theory Ramifications, 26(2):1740012, 28, 2017*.

[FL17] S. Friedl and W. Lück. Universal L2-torsion, polytopes and applications to 3-manifolds. *Proc. Lond. Math. Soc. (3), 114(6):1114–1151, 2017*.

[Ham13] U. Hamenstädt. Bowen’s construction for the Teichmüller flow. *J. Mod. Dyn., 7(4):489–526, 2013*.

[Ham14] U. Hamenstädt. Lines of minima in outer space.* Duke Math. J., 163(4):733–776, 2014*.

[Huy12] D. Huybrechts. A global Torelli theorem for hyperkähler manifolds [after M. Verbitsky].* Astérisque, 348:375–403, 2012. Séminaire Bourbaki: Vol. 2010/2011, Exp. No. 1040*.

[KKP16] S. Katz, A. Klemm, and R. Pandharipande. On the motivic stable pairs invariants of K3 surfaces. In K3 surfaces and their moduli, *volume 315 of Progr. Math., pages 111–146. Birkhäuser/Springer, Cham, 2016*. With an appendix by R. P. Thomas.

[Lüc16] W. Lück. Approximating L2-invariants by their classical counterparts. *EMS Surv. Math. Sci., 3(2):269–344, 2016*.

[Lüc17] W. Lück. Twisting L2-invariants with finite-dimensional representations.* J. Topol. Anal., 2017. DOI:10.1142/S1793525318500279*.

[OS14] S.-i. Ohta and K.-T. Sturm. Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds. *Adv. Math., 252:429–448, 2014*.

[ST14] R. Schneiderman and P. Teichner. Pulling apart 2-spheres in 4-manifolds. *Doc. Math., 19:941–992, 2014*.