Questions tagged [manifolds]
A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
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Is there a discrete lattice analogue of conformal transformations?
There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...
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Homeo-Fixed point property
Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
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Self homeomorphisms of $S^2\times S^2$
Every matrix $A\in SL_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on $\...
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Roadmap for L-Theory
Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the ...
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What is the Status of Borel conjecture today?
Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.
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Stable normal bundle of a manifold
Hi,
in bordism-theory and many bordering areas one has the following construction: Given a manifold M (say closed for the purposes of this discussion and k-dimensional), we embed it into some $\...
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Canonical Metric on Grassmann Manifold
I was curious and quite clueless as to how we can equip the Grassmann Manifold with a canonical metric - I have yet to find anything upon this subject.
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Critical dimensions D for "smooth manifolds iff triangulable manifolds"
I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
For ...
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Cardinality of connected manifolds
Consider the assertion:
Every connected, but not necessarily paracompact, n-manifold is of cardinality
$2^{\aleph_0}$ (at least assuming the axiom of choice).
For n=1 this may be proved via ...
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Can we embed a closed manifold into a homotopy equivalent CW complex?
Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (...
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Is there a closed manifold whose universal cover is $\mathbb{R}^n\setminus\{x_1, \dots, x_k\}$ for some $k > 1$?
There are many closed manifolds with universal cover homotopy equivalent to $\mathbb{R}^n$, they are precisely the closed aspherical manifolds. There are also many closed smooth manifolds with ...
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Volume of an ideal simplex in $\mathbb{H}^3$, idea/intuition behind result
There is the following result on page 160 of Thurston's book "The Geometry and Topology of Three-Manifolds", as follows.
The volume of an ideal simplex in $\mathbb{H}^3$ with dihedral angles $\...
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Are different categories of manifolds non-equivalent (as abstract categories)?
Consider, for instance, the categories of $C^k$-manifolds, where $k=0,1,2,...,\infty,\omega$. ($C^\omega$ means real analytic.) Are these categories pairwise non-equivalent?
Of course, the obviuos ...
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Do $\mathbb{HP}^2\#\overline{\mathbb{HP}^2}$ and $\mathbb{OP}^2\#\overline{\mathbb{OP}^2}$ arise as sphere bundles over spheres?
Recall that $\mathbb{RP}^2\#\mathbb{RP}^2$ is the Klein bottle and can be seen as a non-trivial $S^1$-bundle over $S^1$. In particular, it is the total space of the sphere bundle of $\gamma\oplus\...
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orbit space of a topological manifold
Given a compact Lie group G acting freely on a topological manifold M, is it true that the orbit space M/G is also a topological manifold? If so, why?
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Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness
Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question:
Is it true that one can find a manifold $M$ which is homotopy ...
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Meaning of orientation/orientability over rings other than the integers
This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately.
We consider the homology definition of an orientation for a manifold, as ...
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Existence of normal microbundles
In the same paper where Milnor introduced the concept of microbundles, he gave the following definition. $M$ has a microbundle neighborhood in $N$ if there is a neighborhood $U$ of $M$ in $N$ and a ...
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Lower bounds for Betti numbers of a manifold given its boundary?
Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is ...
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Algebraic properties of the algebra of continuous functions on a manifold.
Does the algebra of continuous
functions from a compact manifold to
$\mathbb{C}$ satisfy any specific
algebraic property?
I'm not sure what kind of algebraic property I expect, but I feel that ...
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Why are there spikes in the number of exotic $n$-spheres for $n \equiv 3 \!\pmod 4$?
Looking at Kervaire and Milnor's1 classification of exotic spheres, for each $n$ there are a finite number of distinct (up to diffeomorphism) smooth structures you can impose on the $n$-sphere (OEIS ...
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Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n = \dim M$
For a connected $n$-manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a point $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\}$. Of course ...
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Detecting topology change of tubular neighbourhoods via smoothness of volume function
Let $M$ be an embedded closed manifold in $\mathbb R^n$, define $M_r=\{x\in\mathbb R^n:d(x,M)<r\}$.
Define $r\in\mathcal S_M$ iff $M_r\subset M_{r+\epsilon}$ is not a homotopy equivalence for all ...
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"The TOP h-cobordism theorem without surgery??"
Kirby and Siebenmann's book on topological manifolds contains the following intriguing passage on page 141:
I believe no such proof has been discovered, though I'd be happy to be corrected on that.
...
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Is there a closed non-smoothable 4-manifold with zero Euler characteristic?
I will just repeat the title:
Is there a closed non-smoothable 4-manifold with zero Euler
characteristic?
I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.
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Elliptic regularity on compact manifold without boundary
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...
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rational homotopy of a manifold
Given a finite dim rational homotopy type satisfying Poincaré duality,
what is the best reference to when it is the rational homotopy type of a fin dim manifold?
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When is the connected sum of manifolds orientation-independent?
Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed?
If $N$ ...
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From Gassmann-Sunada triples to isospectral manifolds
A Gassmann-Sunada triple is a triple $(U,V,W)$ of groups, with $V, W$ subgroups of $U$, such that $U$ and $V$ meet every conjugacy class in $U$ in the same number of elements, and such that $V$ and $W$...
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Non-triangulable 4-manifold as a boundary of some 5 manifold
We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold.
Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...
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Dual cell structures on manifolds
Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $...
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A senseful meaning of 'approximation of manifolds'?
Any continuous function can be uniformly approximated by smooth functions.
I would like to have something similar - in what-ever sense - for continuous manifolds.
For example, by Whitney's theorem, ...
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Is every open topological $d$-manifold homotopy equivalent to a CW-complex of dimension $\leq d-1$?
Let $M$ be a connected open topological $d$-manifold (without boundary).
Whitehead showed that if $M$ has a PL structure, there exists a subcomplex of dimension $\leq d-1$ onto which $M$ deformation ...
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Do colimits of manifolds coincide with underlying colimits as topological spaces?
Categories of manifolds (possibly with extra structure) tend not to have all colimits.
Other questions have addressed when colimits of manifolds exist.
I'd like to know what we can say in general ...
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Which manifolds are sensitive to the cocycle in the Dijkgraaf-Witten model?
Often, TQFTs are defined in families, parametrised by some algebraic data. For example, the Turaev-Viro-Barrett-Westbury TQFTs are parametrised by spherical fusion categories, the Crane-Yetter TQFTs ...
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Reference request - Fibrations between spaces of embeddings
This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...
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A piecewise-linear or topological Fulton-MacPherson compactification
The Fulton-MacPherson compactifications of configuration spaces are smooth manifolds with corners which have the ordered configuration spaces of distinct points in a smooth manifold as their interior. ...
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Elements of infinite order in the topological mapping class group
Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(...
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Simplicial volume
Is there a finite dimensional closed manifold $M$ which is a $K(\pi,1)$, whose fundamental group is not word-hyperbolic, but which has a positive simplicial volume (ie "Gromov norm")?
(Added:) The ...
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Are there invariants of cell complexes similar to the Euler characteristic?
The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely
\begin{equation}
\chi=\...
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Manifolds with rectifiable curves
To begin with, observe that the notion of a rectifiable curve makes sense in, say, a smooth or a PL-manifold but not in merely a topological manifold. Indeed if $f:[0,1]\rightarrow U\subset{\Bbb R}^n$...
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Simplifying triangulations of 3-manifolds
Throughout, by finite triangulation I mean a triangulation consisting of a finite number of triangles.
Suppose $T$ and $T'$ are finite triangulations of a 3-manifold $M$. We will say that $T'$ is ...
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Generators and relations for the 2-dimensional unoriented cobordism category
It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of ...
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Is there a version of the Poincaré–Hopf theorem for manifold with corners?
As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...
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Which topological manifolds do not correspond to strongly Hausdorff locales?
I'm toying with the idea of using locales as a way to define topological manifolds without beginning with points, largely for philosophical reasons.
In this context I think I want to redefine a ...
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Four manifold without point homotopy equivalent to wedge of two-spheres?
Let $X$ be a closed, simply-connected four-manifold. Let $X'$ be obtained from $X$ by removing a point. Is $X'$ homotopy equivalent to a wedge of $S^2$s?
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Can an action of a compact Lie group be nontrivial if it is trivial on the boundary?
Let $G$ be a compact Lie group acting on a connected topological manifold $M$ with boundary. Suppose the action on one boundary component is trivial. Does it follow that the action on the whole of $M$ ...
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Finite domination and Poincaré duality spaces
Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...
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Characteristic polynomials of trees and E8
In thinking about constructing manifolds via surgery or plumbing, the following combinatorial problem comes up:
If T is a tree with adjacency matrix A and I is the identity matrix of the same order, ...
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How does the Lefschetz-Poincare dual torsion linking pairing on manifolds with boundary interact with the maps of the long exact sequence of the manifold-boundary pair?
I'm wondering if anyone can point me to a reference on how the various
Lefschetz-Poincare dual torsion pairings of a manifold with boundary fit
together.
To explain in more detail, consider a ...