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6
votes
1answer
469 views

fundamental domain of universal covering

Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset $D\subset ...
6
votes
0answers
472 views

Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between two Riemmanian manifolds $M_1$ and $M_2$ such that, for each $x_1 \in M_1$, the (codimension-1) measure of the set of points at distance $d$ from $x_1$ ...
9
votes
1answer
303 views

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 $M$ ...
1
vote
1answer
290 views

Do bistellar flips preserve shellability?

I notice there is a strong connection between shellability of simplicial complexes and bistellar flips on these complexes; in particular, adding in a new facet of a shelling induces a bistellar flip ...
1
vote
4answers
2k views

Background to learn about manifolds

Greetings As a necessity to go forward with physics, I find myself in the need to learn about manifolds. Being an engineering student, I don't have the chance to study topology in all its glory. So, ...
3
votes
2answers
354 views

uniqueness of regular/tubular neighborhood with equivariant boundary

Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is ...
0
votes
1answer
253 views

Numbers associated with boundaries of manifolds

I don't know what name if any is attached to the numbers I'm about to describe. For a line segment, [a,b] the number is 1 if for any k in (a,b) and 2 if k=a or k=b. For a square, [a,b] ...
7
votes
0answers
116 views

a “homological dimension” for embedding of manifolds

Let $A\to B$ be a surjective map of commutative $k$-algebras, and suppose $C\to B$ is a free resolution of $B$ as an $A$-algebra, meaning that $C$ is a free non-negatively graded commutative ...
7
votes
2answers
314 views

Fragmenting a homeomorphism of a compact manifold

Let $M$ be a compact manifold and let $f : M \rightarrow M$ be a homeomorphism which is isotopic to the identity. We will say that $f$ can be fragmented if it satisfies the following property. Let ...
7
votes
1answer
479 views

Riemannian metrics on non-paracompact manifolds

After proving the existence of Riemannian metrics on manifolds, one of the students asked if the "paracompactness" is necessary. Of course the standard proof with the partition of unity uses this ...
20
votes
3answers
1k views

The third axiom in the definition of (infinite-dimensional) vector bundles: why?

Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
4
votes
1answer
773 views

Does a Trivial Tangent Bundle Induce a Multiplication?

Let $M$ be a connected smooth manifold, and assume that it is parallelisable; that is, its tangent bundle is trivial. Does $M$ admit an H space structure? That is, does there exist a smooth map ...
8
votes
2answers
726 views

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 ...
7
votes
4answers
877 views

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, ...
4
votes
1answer
1k views

transformation properties of divergence (of a vector field)

Hi, If I have a divergence free vector field defined on a smooth manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an ...
5
votes
2answers
496 views

Definition of the Kervaire invariant for normal maps (as in Browder's book)

Browder's book "Surgery on simply-connected manifolds" defines the Kervaire invariant in a very general setting. My question is: how does one get the more usual definition of the invariant for a ...
15
votes
5answers
1k views

Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?

I'm curious about the following: Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$? Thanks. EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...
12
votes
1answer
1k views

Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)

A celebrated theorem of Rohlin states the following An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero. Simple homological arguments ...
54
votes
16answers
7k views

Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence: "The unorientable surfaces are never discussed ...
5
votes
0answers
179 views

Is there a Whitney-type theorem Cauchy manifolds?

Let $M$ be a Cauchy space whose induced topological space is a second-countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^m$. Does it follow that there exists a subspace $N$ of ...
0
votes
1answer
221 views

What is the topology of the structure group of a fiber bundle?

I dont know how can we topologize the structure group G of a fiber bundle P:E\rightarrow B by transition functions \psi_i^j (Osbern) Do you know an easy and fundamental book on fiber and vector ...
63
votes
4answers
4k views

Which manifolds are homeomorphic to simplicial complexes?

This question is only motivated by curiousity; I don't know a lot about manifold topology. Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. ...
1
vote
1answer
381 views

Understanding manifold GL+(3,R)/SO(3) ?

I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation ...
12
votes
2answers
436 views

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 ...
19
votes
2answers
1k views

Euler Characteristic of a manifold with non-vanishing vector field,

A friend of mine recently asked me if I knew any simple, conceptual argument (even one that is perhaps only heuristic) to show that if a triangulated manifold has a non-vanishing vector field, then ...
8
votes
2answers
694 views

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 ...
5
votes
4answers
909 views

Higher-dimensional braid group?

Let $\Delta$ be 2-disk. Let $C(\Delta;n)$ be a configuration space. i.e.) $C(\Delta;n)= \lbrace (z_1,\ldots,z_n)\in \Delta\times\ldots\Delta | z_i\neq z_j ~\textrm{if}~ i\neq j \rbrace $ Then, it ...
22
votes
3answers
559 views

Ring of closed manifolds modulo fiber bundles

Let $R$ be the ring which is generated by homeomorphism classes $[M]$ of compact closed manifolds (of arbitrary dimension) subject to the relations that $$[F]\cdot [B] = [E]$$ if there exists a fibre ...
5
votes
5answers
769 views

topologically homogeneous space?

I want to know the example which satisfies the following. X is topological space. for every point x,y in X, there exist open nbhd Ux,Uy of x,y which are homeomorphic. X has some kind of good ...
23
votes
2answers
938 views

A Pachner complex for triangulated manifolds

A theorem of Pachner's states that if two triangulated PL-manifolds are PL-homeomorphic, the two triangulations are related via a finite sequence of moves, nowadays called "Pachner moves". A ...
21
votes
2answers
2k views

Are topological manifolds homotopy equivalent to smooth manifolds?

There exist topological manifolds which don't admit a smooth structure in dimensions > 3, but I haven't seen much discussion on homotopy type. It seems much more reasonable that we can find a smooth ...
20
votes
7answers
3k views

Is there a Whitney Embedding Theorem for non-smooth manifolds?

For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can ...
6
votes
2answers
767 views

G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...
7
votes
1answer
475 views

Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions

The Hilbert-Smith conjecture states that If $G$ is a locally compact group which acts effectively on a connected manifold as a topological transformation group then is $G$ a Lie group. It was ...
23
votes
1answer
636 views

“Affine communication” for topological manifolds

There is a situation that comes up regularly in algebraic topology when giving proofs of facts about manifolds, like Poincare duality and the like. The typical sequence goes like this: Prove ...
1
vote
2answers
609 views

Is the complement of a strong deformation retract of a manifold M homotopic equivalent with the boundary of M?

It seems an easy problem but I couldn't prove it. Let $M$ be a manifold with boundary and $N\subset \mathrm{int}(M)$ is a strong deformation retract of $M$. Then I wonder whether $M-N$ is homotopic ...
7
votes
1answer
538 views

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 ...
7
votes
1answer
524 views

Counting submanifolds of the plane

After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane. My ...
9
votes
1answer
1k views

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 ...
17
votes
1answer
1k views

Classification of 1-dimensional manifolds (not second-countable)

It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the ...
13
votes
3answers
1k views

Can we decompose Diff(MxN)?

If you have two manifolds $M^m$ and $N^n$, how does one / can one decompose the diffeomorphisms $\text{Diff}(M\times N)$ in terms of $\text{Diff}(M)$ and $\text{Diff}(N)$? Is there anything we can say ...
5
votes
1answer
637 views

Spins as tensor fields

I have often come across this implicit translation of the classical field of a particle of a given spin into a specific tensor field. But I could not locate any literature from which I could learn ...
59
votes
8answers
7k views

What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The ...
4
votes
2answers
1k views

Poincaré Theorem on presentation from a fundamental polyhedra

Poincaré Theorem on Kleinian groups (groups acting discontinously on Euclidean or hyperbolic spaces or on spheres) provides a method to obtain a presentation of a Kleinian group from a fundamental ...
15
votes
2answers
1k views

A book on locally ringed spaces?

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a ...
10
votes
2answers
737 views

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 ...
4
votes
2answers
409 views

Help me understand boundary terms in actions over nontrivial manifolds

So I have this manifold $M$, along with a metric $g_{\mu\nu}(x)$ and metric-compatible covariant derivative $\nabla_\mu$ (which is not necessarily the one corresponding to the Levi--Civita ...
16
votes
15answers
3k views

Important results that use infinite-dimensional manifolds?

Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...
2
votes
1answer
402 views

Special map from a manifold to GL_n(R)?

I've just finished my first course in differential geometry, so forgive me if this is maybe a silly or well-known question, but given any, say, diffeomorphism of $n$-manifolds $\phi:M\rightarrow N$, I ...
21
votes
3answers
1k views

What manifolds are bounded by RP^odd?

Real projective spaces ℝPn have ℤ/2 cohomology rings ℤ/2[x]/(xn+1) and total Stiefel-Whitney class (1+x)n+1 which is 1 when n is odd, so it follows that odd dimensional ones are boundaries of compact ...