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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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
3answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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)-(...