# Questions tagged [smooth-manifolds]

Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

**13**

votes

**1**answer

239 views

### Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?

Summary
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the ...

**4**

votes

**1**answer

174 views

### Homotopy type of smooth manifolds with boundary

It seems very likely to me that every smooth connected $n$-dimensional manifold with non-empty boundary has the homotopy type of a $(n-1)$-dimensional CW complex. Is that true and how to prove it? (...

**8**

votes

**1**answer

362 views

### 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
...

**9**

votes

**1**answer

327 views

### An almost complex structure on $S^2\times …\times S^2 / \mathbb{Z_2}$

Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...

**2**

votes

**1**answer

139 views

### What is the topological/smooth analogue of Nagata compactification

A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a ...

**3**

votes

**1**answer

99 views

### Making immersions from immersion conjecture into embeddings

Is it true that any smooth manifold of dimension $n$ can be smoothly embedded into $\mathbb{R}^{2n+1-a(n)}$ where $a(n)$ is the number of appearances of digit "1" in the binary expansion of $n$?

**9**

votes

**0**answers

137 views

### An equivalent definition for $\text{Spin}^c$-structures

I'm interested in proving the following proposition ([G], Remark page 48):
Prop: A $\text{Spin}^c$-structure over an oriented vector bundle is equivalent (after stabilizing if the fiber dimension ...

**3**

votes

**0**answers

71 views

### Generic properties of Jacobians of smooth functions

Let $f = (f_1, \dotso, f_n):\mathbb{R}^n \to \mathbb{R}^n$ be a smooth map and let $J$ be its Jacobian (determinant of the matrix with $ij$-th entry $\partial_i f_j$). We introduce the zero sets of $J$...

**5**

votes

**2**answers

164 views

### Equivalence of two definitions of jets of smooth functions

In the literature I have encountered two different definitions of jets of smooth functions, and I was wondering how one could identify these definitions.
One definition is the often encountered ...

**5**

votes

**0**answers

230 views

### Applications of E8 manifold

The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
...

**1**

vote

**0**answers

79 views

### size of local strong stable manifold is measurable

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...

**2**

votes

**0**answers

133 views

### A geometric rank of Riemannian manifolds

There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...

**8**

votes

**0**answers

101 views

### Relating bordism generators in d and d+2 dimensions — an explicit example

This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...

**4**

votes

**0**answers

66 views

### Relating bordism invairants in $d$ and $d+2$ dimensions

Are there some relationship between mapping the bordism invairants of eq.1 and eq.2?
$$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$
and
$$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \...

**5**

votes

**1**answer

192 views

### Manifold generators of O-bordism invariants

If I understand correctly, I can obtain the $O$-cobordism group of
$$
\Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4,
$$
The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...

**4**

votes

**1**answer

125 views

### Nonlinear sigma models with non-compact groups / target spaces

A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.
The target manifold T is equipped with a Riemannian metric g. Σ is ...

**2**

votes

**0**answers

97 views

### The isometry groups of flag manifolds

For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ ...

**4**

votes

**1**answer

287 views

### (Co)bordism invariant of Eilenberg–MacLane space becomes vanished

Consider a (co)bordism invariant
$$
u_2 Sq^1 u_2+Sq^2 Sq^1 u_2
$$
obtained from
$$
\Omega^5_{O}(K(\mathbb{Z}/2,2)).
$$
Here $u \in H^2(K(\mathbb{Z}/2,2),\mathbb{Z}_2)$. The $K(\mathbb{Z}/2,2)$ is ...

**3**

votes

**1**answer

182 views

### Is there a vector field such that one differential form is the Lie derivative of the other?

I'm looking for a reference or answer for the following question:
Let $M$ be an (compact and orientable, if it helps) smooth manifold and $\nu$ and $\mu$ two differential forms. I'm looking for ...

**2**

votes

**1**answer

203 views

### A question on eversion of (odd) spheres

At the right column of the page 654 of the paper,
R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS it is written "There can be no eversion of ...

**2**

votes

**0**answers

80 views

### The complex Clifford algebra

If $(E,g,w)$ is a vector space $E$ with a metric $g$ and a symplectic form $w$; then we can define the complex parts $(1,0)$ and $(0,1)$, so that the complex Clifford algebra is: $$e_1 . f_1 + f_1 . ...

**8**

votes

**3**answers

907 views

### 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.

**1**

vote

**0**answers

43 views

### Upper bounds on $\epsilon$-covers of arbitrary compact manifolds

Let $M \subset \mathbb{R}^d$ be a compact $k$-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 ...

**3**

votes

**1**answer

218 views

### a counter-example of Lie Group

Is there an example of a $n$-dimensional Lie group whose left invariant $n$-form is not right invariant? As far as I know, the Lie group can't be compact. But I don't know how to construct an example.

**4**

votes

**0**answers

89 views

### Lower bound on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-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 $...

**3**

votes

**1**answer

119 views

### Special spheres: principal curvatures with different signs

For $\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface $M^n$ of a riemannian manifold $(N^{n+1},g)$ is $\varepsilon$-convex whenever all principal curvatures of $...

**3**

votes

**1**answer

52 views

### Cut on hypersurfaces and angular defects

I like very much the elementary property that if one cuts a geodesic triangle onto a sphere (one can use 3 plans that contain $0$). The cut surface of the sphere is given by the sum of the angles of ...

**13**

votes

**1**answer

267 views

### Handlebody decomposition of 4-spheres without 3-handles

There used to be many candidates for an exotic 4-sphere, but a lot of them are now known to be the standard smooth $S^4$. The ones of Cappell-Shaneson (maybe not all of them?) were described in terms ...

**3**

votes

**0**answers

82 views

### Trying to understand why this local coordinates parametrizes a manifold

First of all, I would like to say that I think this question fits better on Math Overflow than on Math Stack Exchange, in view of the proposal of the two sites. However, if my analysis of the ...

**2**

votes

**0**answers

116 views

### Classification of Principal $G$ bundles and vector bundles in smooth sense

Suppose $G$ is a Topological group then classification theorem of Principal $G$ bundles says that
there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a ...

**3**

votes

**0**answers

92 views

### Cobordant of 5d manifolds, and the generalization of bordisms

Some of the 5-dimensional manifolds are (co)bordant via oriented cobordism.
For example, if I understand correctly, 5-dimensional Dold manifold and Wu manifold are manifolds which are cobordant to ...

**6**

votes

**0**answers

197 views

### Quotient space, a fundamental group, and higher homotopy groups 2

Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...

**4**

votes

**0**answers

258 views

### Non-Abelian fundamental group? — a bizarre example

For the quotient space $G=G_0/G_1$, knowing the homotopy
groups of $G_0$ and $G_1$, one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(G_1) \to \pi_n(G_0) \to \...

**5**

votes

**0**answers

151 views

### Are there non-cuspy triangulations of smooth manifolds?

In (as it turned out my misunderstanding of) the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a ...

**7**

votes

**0**answers

161 views

### Cell structures of Dold manifold and Wu manifold

In Dold's 1956 paper Erzeugende der Thomschen Algebra N, Dold studied the Dold manifold $P(m,n)=(S^m\times\mathbb{CP}^n)/\tau$ where $\tau$ acts as $-1$ on $S^m$ and a complex conjugation on $\mathbb{...

**1**

vote

**0**answers

58 views

### Construction of homogeneous space

Given a Hilbertspace $\mathcal{H}$ of dimension $n<\infty$ and all hermitian matricies $Symm(\mathcal{H})$. I'd guess that the set $M_{2,2} \subset Symm(\mathcal{H})$ of all matricies of rank 4 and ...

**4**

votes

**1**answer

139 views

### Surjections on generalized homology theory

In Davis-Januszkiewica´s paper Hyperbolization of polyhedra , the authors hyperbolized every closed n-manifolds K to get a new manifold, say M(K),together with a map $f_K$ from M(K) to K, then they ...

**1**

vote

**0**answers

115 views

### Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]

[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...

**1**

vote

**0**answers

73 views

### Injective differential of linear operators on a Hilbertspace

Given a complex Hilbertspace $\mathcal{H}$ of dimension $\dim(\mathcal{H}) = d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\vert\quad \text{rank}(q) = 4 \quad \wedge \lambda^q_{1,2} < 0\ \ \...

**17**

votes

**1**answer

867 views

### What are the possible Stiefel-Whitney numbers of a five-manifold?

On a compact five-manifold, the Stiefel-Whitney number $w_2w_3$ can be nonzero.
An example is the manifold $SU(3)/SO(3)$, and also another example is a $\mathbb{CP}^2$ bundle over a circle where the ...

**2**

votes

**3**answers

332 views

### Classification of line bundles by second cohomology of a manifold

In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.
$H^2(M,\mathbb{Z}...

**4**

votes

**0**answers

94 views

### $\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities

Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...

**10**

votes

**2**answers

397 views

### Is the boundary of a manifold topologically unique? [duplicate]

Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \...

**3**

votes

**0**answers

113 views

### Diffeomorphism group action on the space of embeddings

Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...

**2**

votes

**0**answers

133 views

### Quotient by a non-free action of a Lie group and manifolds with corners

The quotient manifold theorem says that
If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$...

**4**

votes

**0**answers

239 views

### A cohomology associated to a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold.
Let $$\Omega_{\omega}^k(M)=\{\alpha \in \Omega^{k}(M)\mid \alpha \wedge \omega \;\;\text{is an exact form}\}$$
Then we have a chain comlex$$\...

**3**

votes

**0**answers

81 views

### Fenchel conjugate on a Hadamard manifold

Let $M$ be a Hadamard manifold and let $F:M\to\mathbb{R}$ be a real-valued convex function on $M$. What would be the Fenchel-Young conjugate of $F$?
In general for a real locally convex vector space $...

**1**

vote

**1**answer

123 views

### Lie brackets of automorphisms

Let $F$ be the vector fields of a differential manifold $M$, let $[X,Y]$ be the Lie brackets of $F$, now let $a$ be an automorphism of $F$ for the structure of real vector space of $F$. I consider now ...

**7**

votes

**1**answer

226 views

### Thom's first isotopy lemma

Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that
$f|_S:S\to P$ is proper and $f|_X:X\to P$ is a ...

**1**

vote

**1**answer

206 views

### Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset
$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$
is a manifold of dimension $2n(2r)-(...