Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.

  • Cobordism Theory for DIFF/Differentiable/smooth manifolds

However, there are Topological Manifolds which are not Differentiable Manifolds.

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So my question here for experts is that what do I need to beware and pay attention in order to master a cobordism theory of Topological Manifolds? What are the main differences of the computations of the bordism groups for the given following structures:

Say,

  1. Cobordism Theory of TOP/topological manifolds

  2. Cobordism Theory for PDIFF/piecewise differentiable manifolds

  3. Cobordism Theory for PL/piecewise-linear manifolds

p.s. Are there Spin, Pin$ ^+$ , and Pin$ ^-$ versions of these cobordism theories of Topological Manifolds computed in the literature explicitly?

Is the injectivity radius (semi) continuous on a non-complete Riemannian manifolds?

Let $ \mathcal{M}$ be a Riemannian manifold, and let $ \mathrm{inj} \colon \cal M \to (0, \infty]$ be its injectivity radius function.

It is known that if $ \cal M$ is connected and complete, then $ \mathrm{inj}$ is a continuous function: see for example [Lee, Introduction to Riemannian Manifolds, 2018, Prop. 10.37].

What is known in the case where $ \cal M$ is not complete? Is $ \mathrm{inj}$ also continuous? If not, is there a known counter-example? Would $ \mathrm{inj}$ still be semi-continuous?

This question is similar to the question “The continuity of Injectivity radius”, but the discussion there focuses on compact or complete manifolds.

Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?

Is it always possible to write a derived manifold as a homotopy colimit of principal derived manifolds (i.e. zero sets of smooth functions)? This is true for schemes and derived schemes, so it seems likely to be the case as principal derived manifolds are “affine” objects.

Do (3+1)-dimensional Lorentzian manifolds admit unique smoothings?

Of course, 3-dimensional topological manifolds admit unique smoothings while 4-dimensional topological manifolds generally do not. A (3+1)-dimensional topological Lorentzian manifold (definition below) is a 4-manifold, but with extra structure which locally fibers it over 3-dimensional manifolds, so perhaps there’s some hope that such a space can be uniquely smoothed.

I’m not sure there’s a standard definition of a topological Lorentzian manifold, so here’s a stab at it. Let $ n \in \mathbb N$ .

Definition: Let $ X$ be a topological space of dimension $ n+1$ .

  • The sheaf of $ X$ -trajectories is the sheaf $ C^X: O(\mathbb R)^{op} \to Set$ on the topological space $ \mathbb R$ defined by $ C^X(U \subseteq \mathbb R) = Map(U, X)$ .

  • A pre-causal structure on $ X$ consists of a subsheaf $ T^X \subseteq C^X$ of weakly timelike $ X$ -trajectories. An isomorphism of pre-causal structures is a homeomorphism of spaces and a commuting isomorphism of sheaves.

Note that Minkowski space $ \mathbb R^{n+1}$ has a canonical pre-casual structure, where we say that a path is weakly timelike if it is locally uniformly approximable by smooth weakly timelike paths (where in the smooth setting “weakly timelike means the tangent vector is always timelike or lightlike).

  • If $ X$ is a topological manifold of dimension $ n+1$ , a topological $ (n+1)$ Lorentzian structure on $ X$ consists of a pre-causal structure on $ X$ such that $ X$ admits an open cover of subspaces one which the pre-causal structure is isomorphic to Minkowski space $ \mathbb R^{n+1}$ .

Note that a smooth Lorentzian manifold has canonically the structure of a topological Lorentzian manifold.

Question: Does every topological Lorentzian manifold of dimension 3+1 arise uniquely as the underlying topological Lorentzian structure on a smooth Lorentzian manifold?

Covering manifolds with some other manifolds

Let $ M$ ,$ N$ be $ n$ -dimensional manifolds. Let $ D_{1},D_{2},\dots D_{k} $ be $ n$ -dimensional manifolds embedded in $ M$ and $ \cup_{i=1}^kD_{i}=M$ and each $ D_i$ is homeomorphic to $ N$ . Question is following.

Problem Consider all embedding of $ \cup_{i=1}^kD_{i}$ . Determine the minimum of the value $ k$ .

Example When we consider $ M$ is 2-dimensional torus and each $ D_i$ is a 2-dimensional disk, the minimum value is 3. I thought $ M=L(5,2)$ and each $ D_i$ is a 3-dimensional ball, and I couldn’t determine the minimum.

Can you answer?

Reference request: Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann’s approach)

This question is a cross post from Math.SE. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this related question, but in my opinion it is not a duplicate from mine.

I was reading about geometry in metric spaces from different books, two of them are: (1) A course in metric geometry by Y. Burago, D. Burago and S. Ivanov; and (2) Metric spaces of non-positive curvature by M. Bridson and A. Häfliger. Both develop the Alexandrov’s approach to curvature, which uses comparison triangles with the constant curvature model spaces.

For a normed space $ X$ , the following statements are equivalent:

  1. $ X$ has curvature $ \leq\kappa$ in Alexandrov’s sense, for some real number $ \kappa$ .
  2. $ X$ has curvature $ \leq 0$ in Alexandrov’s sense.
  3. The norm on $ X$ is induced by an inner product.

So it seems to me that Alexandrov’s approach is not very informative in the normed case. On the other hand, a geodesic space has non-positive curvature in the Busemann’s sense if its metric is convex, in general this is a weaker notion than Alexandrov’s, and in the normed case the following statements are equivalent:

  1. $ X$ has non-positive curvature in the Busemann’s sense.
  2. $ X$ is uniquely geodesic, that is, every pair of points is joined by a unique geodesic (the linear segment between them).
  3. $ X$ is strinctly convex, that is, the ball in $ X$ is strictly convex which means that for every pair of different vectors $ v$ and $ w$ of norm equal to $ 1$ we have that $ tv+(1-t)w$ has norm strictly less than $ 1$ for every $ t$ in $ (0,1)$ .

So it seems to me that this weaker notion is the appropriate notion for non-positive curvature in the normed case and I think also for finsler manifolds. I have never studied finsler geometry, but I am very interested in studying metric geometry from this approach. And I do not know where I should start.

My question is: What is a good introductory book about finsler manifolds from the metric geometry point of view? What is a good introductory book for the Busemann’s approach? If there was not an introductory book available, a reference to an advanced one along with references that cover the necessary background would be very welcome.

In Math.SE, user @HK Lee has suggested the paper On intrinsic geometry of surfaces in normed spaces by D. Burago and S. Ivanov. And I have found the following references, although I need the advice of the experts:

  1. An introductory textbook by A. Papadopoulos about the Busemann’s approach: Metric Spaces, convexity and non-positive curvature.

  2. A textbook by H. Busemann: The geometry of geodesics

  3. Two interesting papers by H. Busemann: The geometry of finsler spaces and Spaces with non-positive curvature.

Thanks in advance!

What is the standard smooth structure of a disjoint union of smooth manifolds?

I am following Lee in his introduction to smooth manifolds page 442. He explains that if $ \{M_j\}_{j = 1}^\infty$ is a countable collection of smooth $ n$ -manifolds with or without boundary and let $ M = \coprod_{j=1}^\infty M_j$ , then there is an isomorphism of the de Rham cohomology groups $ H^p(M)$ and $ \prod_{j=1}^\infty H^p(M_j)$ .

But what is the standard smooth structure that turns $ \coprod_{j=1}^\infty M_j$ into a smooth manifold?

Asymptotic Bound on Minimum Epsilon Cover of Arbitrary Manifolds

Let $ M \subset \mathbb{R}^d$ be a compact smooth $ k$ -dimensional manifold embedded in $ \mathbb{R}^d$ . Let $ \mathcal{N}(\epsilon)$ denote the size of the minimum $ \epsilon$ cover $ P$ of $ M$ ; that is for every point $ x \in M$ there exists a $ p \in P$ such that $ \| x – p\|_{2}$ .

Is it the case that $ \mathcal{N}(\epsilon) \in \Theta\left(\frac{1}{\epsilon^k}\right)$ ? If so, is there a reference?

Rationally connected Kahler manifolds are porjective

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:

https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf

She writes in this remark the following:

Remark 0.5 A compact Kahler manifold $ X$ which is rationally connected satisfies $ H^2(X, {\cal O}_X) = 0$ , hence is projective.

I understand that a Kahler manifold with $ H^2(X, {\cal O}_X) = 0$ is projective. However, I don’t understand why a Kahler manifold that is rationally connected has $ H^2(X, {\cal O}_X) = 0$ . Indeed, the definition for rational contectedness that Voisin is using is the following:

Definition 0.3 A compact Kahler manifold $ X$ is rationally connected if for any two points $ x, y\in X$ , there exists a (maybe singular) rational curve $ C\subset X$ with the property that $ x\in C$ , $ y\in C$ .

So my question is the following: How to prove this remark starting with Definition 0.3?