The manifolds tag has no wiki summary.

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### Examples of non-simply connected manifolds with trivial H^1

It is known that, if a topological space is simply connected,its first homology group vanishes. The converse is not true, since for every presentation of a (say, finite) perfect group G we can ...

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### symplectic form with partition on unity

Assume $M$ is a $2n-$dimensional differentiable manifold. Let $(U_{i})$ be a open covering of $M$. With respect to this covering let $\rho_{i}$ be a partition of unity. Assume that on each $U_{i}$ we ...

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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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### A “meta-mathematical principle” of MacPherson

In an appendix to his notes on intersection homology and perverse sheaves, MacPherson writes
Why do we want to consider only spaces $V$ that admit a decomposition into manifolds? The intuitive ...

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### How is the differential in complex cobordism defined?

This is my first MO question...hopefully it's not a bad one...
Background: As a stable homotopy theorist, I like to think of complex cobordism $MU$ as a ring spectrum. If I needed to get my hands ...

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### Question concerning h-cobordisms

Suppose we have a cobordism $W$ of manifolds $M_0$ and $M_1$ and suppose the inclusion of $M_0$ into $W$ is a homotopy equivalence. Is the same true for the inclusion of $M_1$ (ie. is $W$ already an ...

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### cayley transform for non-square matrices

Hi,
I am optimizing a function over a matrix $U$, where $U \in \mathbb{R}^{m \times n}$ and $U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint $U^TU = ...

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### 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 ...

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### 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 $x_1$ ...

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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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### 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 ...

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### 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, ...

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### 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 ...

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**1**answer

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### 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] ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 ...

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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 ...

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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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $n\geq 4$. ...

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### 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 ...

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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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### 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 ...

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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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### “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 ...

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### 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 ...

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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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...