Blockseminar Constructions of Aspherical Manifolds and Complexes

Blockseminar Constructions of aspherical manifolds and complexes

Organizers: Alexander Engel, MichalMarcinkowski, Christopher Wulﬀ Date: 24.-28. September 2018 Place: Frauenchiemsee

Symmetric spaces

The main reference for this part is [10]. Talk 1 (Lie groups and Lie algebras, and the exponential maps). Define Lie groups and Lie algebras, define the adjoint representation ad(−) and the center of a Lie algebra (p. 98–100). Discuss several examples of Lie groups and their Lie algebras (use references of your choice). Give the definition of the exponential map exp(−) from p. 104 (and discuss what you need from p. 102f) and of the exponential map Exp(−) from p. 33. Finally, prove Theorem 1.7 from p. 105 (for this you will have to discuss some things from §6 in Chapter I).

Talk 2 (Semi-simple Lie algebras and the Cartan decomposition). Define the adjoint group Int(−), the automorphism group Aut(−) and the adjoint representation Ad(−) of a Lie group (p. 126f). Define compactly imbedded subalgebras and compact Lie algebras (p. 130). Define Killing forms and semi-simple Lie algebras (p, 131) and prove Propo- sition 6.6 (p. 132). State Proposition 6.10 without proof (p. 135). Give examples of semi-simple Lie algebras (use references of your choice). Discuss complexifications of Lie algebras and real forms (p. 180), state Theorem 6.3 (leave out the proof since we do not discuss roots in this seminar) and discuss Theorem 7.1 (p. 181f). Define Cartan decompositions (p. 183) and conclude from Theorems 6.3 and 7.1 that they always exist for semi-simple Lie algebras over R.

Talk 3 (Symmetric spaces of non-compact type). You have to discuss many parts of §3 in Chapter IV. Start with the definition of a Riemannian globally symmetric space (p. 205), discuss Lemma 3.1 and state Lemma 3.2 (without proof). Discuss from Theorem 3.3 only the things which you need later. Define what a (Riemannian) symmetric pair is (p. 209) and what an orthogonal symmetric Lie algebra is (p. 213), and discuss Proposition 3.6. Discuss Theorem 4.1 (p. 215) and prove finally Theorem 1.1(iii) of Chapter IV.

Aspherical presentation complexes

Talk 4 (Small cancellation and one-relator groups). Prove that the presentation complex of a small cancellation group is aspherical

1 [14, Thm. 1(2)]. For this you will have to relate the deﬁnition of an aspherical presentation from [14] to asphericity of the presentation complex ([9, p. 32]) and show (at least sketch) [14, Lem. 13]. Prove that the presentation complex of a torsion-free one-relator group is aspherical [16, Thm. 7.1].

Reﬂection group trick

Talk 5 (Coxeter groups). Define Coxeter groups by presentation ([3, Def.3.3.2], ignore pre-Coxeter systems). Give examples of Coxeter groups: dihedral groups [3, §3.1], your favourite example from [3, §6.5], discuss the 3-dimensional example given by the tiling of H3 by the right angled dodecahedron ([18, Ex.1.4.5] for existence of such a dodecahedron, if you have doubts about this construction a look at [3, Thm.6.4.3] might be helpful). Define the canonical representation [3, §6.12]. Note that the 3-dimensional example above may be used to construct closed 3-dimensional hyperbolic manifolds (by Selbergs’ lemma every Coxeter group is virtually torsion free). State [3, Lem.4.7.2] and [3, Lem.4.7.3] (these combinatorial lemmas will be needed for the next talk). If time permits give ideas of the proofs. Talk 6 (Basic construction). Define U(G, X) assuming that G = (W, S) is a Coxeter group [3, §5.1]. Discuss the nerve of a Coxeter system [3, §7.1] and the complex U(W, K)[3, §7.2]. (focus on the last paragraph before Thm.7.2.4, you don’t have to introduce abstract simplicial complexes and flags). Sketch the proof of [3, Thm.8.1.2] and [3, Thm.9.1.4]. Talk 7 (Aspherical manifolds not covered by Rn). Proof [3, Thm.10.5.1]. You have to explain in details the crucial Remark 10.5.2. To do that, you need to look at the proof of [3, Thm.9.2.2]. Focus rather on the fundamental group at infinity, not on semi-simplicity. Our examples are one-ended, thus while defining the fundamental group at infinity specify the discussion to the one-ended case. If time permits, give a concrete example of a homology sphere (e.g. [3, Ex.10.3.2]). Talk 8 (Aspherical manifolds not homotopy equivalent to smooth manifolds). Describe the reflection group trick: cover [3, §11.1] without Prop.11.1.3. As examples discuss shortly [3, Ex.11.2.1,Ex.11.2.2]. The main part of the talk is to prove [3, Thm.11.3.2] (see as well [4]). Give background on homotopy theory: explain what is an infinity loop space, classifying space and Spivak fibration (see [3, App.H2] and your favourite book on homotopy theory).

CAT(0) geometry

Talk 9 (CAT(κ) spaces and Gromov’s version of the Cartan–Hadamard theorem). Give the deﬁnition of CAT(κ) spaces ([1, II.1]), discuss examples from [1, II.1.15] and prove [1, Thm. 4.1]. Talk 10 (CAT(0) cube complexes). Deﬁne cubical complexes [1, Thm. I.7.32] and prove Gromov’s link condition for non-positive curvature [1, Thm. II.5.20].

2 Examples: Salvetti complex for right angled Artin groups [2, §3.6], U(W, K) for right angled Coxeter group W [3, §12.2]. If time permits, discuss K(π, 1) conjecture for general Artin groups, [13, Def.4.5] and [13, Def.4.7].

Asphericalization `ala Kan–Thurston

Talk 11 (Theorem of Kan–Thurston). Discuss and prove the Theorem of Kan–Thurston (you can ignore the statements about uniqueness of the spaces and about the Quillen +-construction). You can use either the original paper [11] or the article [12].

Hyperbolisation of polyhedra

The reference for this part is [5]. Talk 12 (The Williams functor). Discuss thoroughly Section 1. Talk 13 (Main theorem and ﬁrst application). Discuss Section 4 and prove the Main Theorem (4a.3). Explain the construction of hyperbolized simplices as needed in the main theorem (this is the main part of Section 4). As an application prove Theorem A from the introduction of the paper. Discuss also Theorem B. Talk 14 (Non-triangulable aspherical manifolds). Discuss Section 5a, i.e., Theorem (5a.1) and Corollary (5a.4). Explain as much as possible about the manifold E8 and the algebraic topology you need in the proof (use references of your choice), such that the audience can follow your exposition. Explain the plumbing construction for the construction of the manifold E8.

More aspherical spaces

Talk 15 (Universal realisation of cycles). Explain Steenrod’s realisation problem, [8, §1] & [6, §1] (you can discuss Sullivan’s resolution of singularities [6, §1] and the idea of the construction [6, §2]). The goal is to prove [8, Thm. 2.5]. If time permits discuss [7, Cor. 1.6]. Talk 16 (Aspherical manifolds and expanders). Sketch the construction of groups containing an isometric copy of an expander [15, Thm. 4] (some deﬁnitions concerning small cancellation theory are explained in Talk 4). Cover [15, §3.2.2] and state [17, Thm. 1.1] without proof.

References

[1] M. R. Bridson and A. Haeﬂiger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. [2] R. Charney. An introduction to right-angled Artin groups. Geom. Dedicata, 125:141–158, 2007.

3 [3] M. W. Davis. The geometry and topology of Coxeter groups, volume 32 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2008.

[4] M. W. Davis and J.-C. Hausmann. Aspherical manifolds without smooth or PL structure. In Algebraic topology (Arcata, CA, 1986), volume 1370 of Lecture Notes in Math., pages 135–142. Springer, Berlin, 1989. [5] M. W. Davis and T. Januszkiewicz. Hyperbolization of polyhedra. J. Diﬀerential Geom., 34(2):347–388, 1991. [6] A. A. Gaifullin. Combinatorial realisation of cycles and small covers. In European Congress of Mathematics, pages 315–330. EMS, 2013. [7] A. A. Gaifullin. Universal realisators for homology classes. Geom. Topol., 17(3):1745–1772, 2013.

[8] A. A. Gaifullin. Small covers over graph-associahedra and realization of cycles. Mat. Sb., 207(11):53–81, 2016. [9] S. M. Gersten, editor. Essays in group theory, volume 8 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1987.

[10] S. Helgason. Diﬀerential Geometry, Lie Groups, and Symmetric Spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, 2001. Reprint of the 1978 edition. [11] D. M. Kan and W. P. Thurston. Every connected space has the homology of a K(π, 1). Topology, 15(3):253–258, 1976.

[12] C. R. F. Maunder. A short proof of a theorem of Kan and Thurston. Bull. London Math. Soc., 13(4):325–327, 1981. [13] J. McCammond. The mysterious geometry of Artin groups. http://web.math.ucsb.edu/~jon.mccammond/papers/mysterious-geometry.pdf. [14] Y. Ollivier. On a small cancellation theorem of Gromov. Bull. Belg. Math. Soc. Simon Stevin, 13(1):75–89, 2006. [15] D. Osajda. Small cancellation labellings of some inﬁnite graphs and appli- cations. arxiv:1406.5015. [16] A. Putman. One-relator groups. https://www3.nd.edu/~andyp/notes/OneRelator.pdf. [17] M. Sapir. A Higman embedding preserving asphericity. J. Amer. Math. Soc., 27(1):1–42, 2014. [18] W. P. Thurston. Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.