TOPOLOGICAL MANIFOLDS GOPAL ANANTHAKRISHNA, ANTHONY CONWAY, DANICA KOSANOVIĆ, CHRISTIAN KREMER, ISACCO NONINO, MARK POWELL, ARUNIMA RAY, AND BENJAMIN RUPPIK Contents 1 Overview .............................................................................. 2 1.1 Outline 4 1.2 Conventions 4 2 Definitions of topological manifolds ...................................................... 5 3 Invariance of domain ................................................................... 8 4 More foundational properties of topological manifolds ...................................... 12 5 Wild embeddings ....................................................................... 13 6 Collars and bicollars .................................................................... 19 7 The Schoenflies theorem ................................................................ 23 7.1 Overview of proof strategy 23 7.2 Whitehead manifold 23 7.3 Shrinking cellular sets 24 7.4 Schoenflies in the smooth category 31 8 Spaces of embeddings and homeomorphisms .............................................. 34 8.1 The compact-open topology on function spaces 34 8.2 Spaces of embeddings and isotopy 35 8.3 Immersions 36 9 Microbundles and topological tangent bundles............................................. 38 9.1 Microbundles 38 9.2 Constructions of microbundles 40 9.3 Normal microbundles 42 9.4 Precursor to smoothing theory 43 9.5 Kister’s theorem 44 10 Homeomorphisms of Rn and the torus trick ............................................... 52 10.1 Homeomorphisms bounded distance from the identity and Alexander isotopies 52 10.2 Torus trick – the proof of the Černavskiˇı-Kirby theorem 53 10.3 Local contractibility 62 11 Piecewise linear manifolds ............................................................... 64 12 Stable homeomorphisms and the Annulus Theorem ........................................ 65 12.1 Stable homeomorphisms 66 12.2 Stable homeomorphism in the smooth and PL categories 70 12.3 Proof of the stable homeomorphism theorem 73 12.4 Consequences of SHn and ACn 75 2020 Mathematics Subject Classification. 57-01, 57-02 (Primary); 57K40, 57N30, 57N35, 57N37, 57N40, 57N45, 57N50, 57N55, 57N60, 57N65, 57N70, 57N75 (Secondary). Date: February 9, 2021. 1 2 GA, AC, DK, CK, IN, MP, AR, AND BR 13 PL homotopy tori ...................................................................... 78 13.1 Classification theorems 78 13.2 The structure set 79 13.3 Normal bordism and the surgery obstruction 79 13.4 Wall realisation and the size of each normal bordism class 81 13.5 Computations of the surgery obstruction maps 82 14 Local contractibility for manifolds and isotopy extension ................................... 89 14.1 Handle decompositions 90 14.2 Handle straightening 90 14.3 Applying handle straightening 93 14.4 Proof of the isotopy extension theorem 95 15 Classifying spaces ...................................................................... 98 15.1 Semi-simplicial sets 98 15.2 Defining classifying spaces 99 15.3 Comparing stable classifying spaces. 99 16 Smoothing and PL-ing theory ........................................................... 100 17 The Homotopy groups of TOP / PL and TOP /O ..........................................104 PL 17.1 Smoothing of piecewise-linear manifolds and the homotopy groups of O 104 TOP 17.2 Homotopy groups of O 105 17.3 The homotopy groups of TOP / PL 105 18 Proof of Concordance implies isotopy.....................................................109 18.1 CAT handle straightening 109 18.2 Proof of concordance implies isotopy 110 19 The Product Structure theorem..........................................................112 19.1 CAT structures on Euclidean spaces 112 19.2 The proof of the product structure theorem 112 Solutions to exercises ...................................................................113 TOPOLOGICAL MANIFOLDS 3 1 Overview We begin with an overview of the field of topological manifold theory in general and a preview of what we will discuss. First, we define topological manifolds. Definition 1.1 (Topological manifold). A topological space M is said to be an n-dimensional topological manifold if it is (i) Hausdorff, i.e. any two points may be separated by open neighbourhoods; (ii) locally Euclidean, i.e. for every x ∈ M there is an open neighbourhood U 3 x that is n n n homeomorphic to either R or R+ := {~y ∈ R | y1 ≥ 0}; and (iii) paracompact, i.e. any open cover has a locally finite refinement. n Note that by allowing the possibility of R+ we are defining what some authors call a “manifold with boundary”. With our definition we avoid having to specify that a boundary is permitted, however this means we must then stipulate when one is expressly forbidden. You may also have seen other definitions of manifolds, e.g. requiring second countability or metrisability. We will see presently that some other definitions are equivalent to the one above, and in the exercises you will explore examples of spaces lacking one or other of the above properties. The word “manifold” comes from German. Specifically, Riemann used the term Mannig- faltigkeit in his PhD thesis to describe a certain generalisation of surfaces. This was translated to “manifoldness” by Clifford. Prior to Riemann, mathematicians had classically studied geometry, first Euclidean, then spherical and hyperbolic. Surfaces were studied in depth, including by Riemann. As you probably know the first systematic account of the field of topology was in Analysis situs by Poincaré, and the first definition he wrote down was of what he called a manifold. In modern terms, he defined a smooth manifold. Here is a quick reminder of the definition (the modern one, not Poincaré’s). Definition 1.2 (Smooth manifold). Let M n be a topological manifold. A chart on M is a pair ∼ = n (U, ϕ) where U ⊆ M is open and ϕ: U −→ R is a homeomorphism. If (U, ϕ) and (V, ψ) are two charts on M such that U ∩ V 6= ∅ then the map ψ ◦ ϕ−1 is said to be a transition map. Transition map are homeomorphisms, since each is a composition of homeomorphisms by definition. If ψ ◦ ϕ−1 is further a diffeomorphism then the charts (U, ϕ) and (V, ψ) are said to be smoothly compatible. A smooth atlas for M is a collection of smoothly compatible charts for M whose domains cover M. A smooth structure on M is a maximal smooth atlas, where maximal means that any chart smoothly compatible with the atlas is already contained in the atlas. A couple of remarks are in order. First, Poincaré’s original definition of a (smooth) manifold had been as a subset of Euclidean space satisfying a given collection of smooth functions. The Whitney embedding theorem from the 1930s showed that every smooth n-manifold (satisfying 2n+1 the definition above) embeds in R , and so the two notions coincide. Second, the definition above indicates a recipe for imposing more structure on topological manifolds. By requiring the transition maps to be smooth, we obtain smooth manifolds. Similarly, by imposing further (or fewer) conditions, e.g. symplectic, complex, C1, etc., we may produce more categories of manifolds. In this course, we will focus on unadulterated topological manifolds, with occasional cameos by smooth manifolds and piecewise-linear manifolds. We define the latter next. As you probably noticed in your algebraic topology courses, it is often convenient to work with simplicial complexes rather than purely abstract spaces, e.g. when computing homology groups. This was especially true in the early days of topology. Definition 1.3. A manifold is said to be triangulated if it is homeomorphic to the geometric realisation of a (locally finite) simplicial complex. 4 GA, AC, DK, CK, IN, MP, AR, AND BR A piecewise-linear manifold, often called a PL manifold is a manifold with a particularly nice triangulation. Definition 1.4 (PL manifold (preliminary)). An n-manifold is piecewise linear (PL) if it has a triangulation such that the link of every vertex is a PL (n − 1)-sphere or PL (n − 1)-ball. Rest assured, we will carefully define what a PL sphere is later in the course. An intuitive way to think about the definition is that it is a strengthening of the “locally Euclidean” condition in the definition of a manifold, specifically that not only does each point have a neighbourhood homeomorphic to Euclidean space, but that such neighbourhoods may further be taken to be PL equivalent to Euclidean space. An alternative definition of PL manifolds requires that the transition maps be piecewise-linear maps on Euclidean space (also to be defined carefully in the future). In other words, a PL manifold is a topological manifold with a maximal PL atlas. A result of Dedecker [Ded62] shows that the two definitions coincide. By definition, both smooth and PL manifolds are topological manifolds, by forgetting the extra structure. By results of Cairns (1934) and Whitehead (1940) every smooth manifold is PL. Since the very inception of manifold theory, e.g. in Analysis situs, there has been much interest in the relationship between these three categories. Here are some other fundamental questions. (1) Is a given CW complex homotopy equivalent to a TOP manifold? PL? DIFF? (2) Given two manifolds, are they homotopy equivalent? Are they homeomorphic? If they are PL or smooth, are they PL homeomorphic or diffeomorphic respectively? (3) When do manifolds embed in one another? (4) For a given topological manifold M, what is the space of self-homeomorphisms Homeo(M)? Given a pair of manifolds M and N, what is the space of embeddings Emb(M, N)? These are huge, very general questions. We will make