**5**

votes

**0**answers

225 views

### Embedding tower in low codimension

If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm{holim} T_k F$.
The ...

**3**

votes

**1**answer

501 views

### Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?

Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$.
Question: Is the space ...

**8**

votes

**1**answer

364 views

### Construction of exotic spheres that do not bound parallelizable manifolds

There are at least two ways to construct homotopy spheres that bound parallelizable manifolds, namely Milnor's plumbing construction and Brieskorn's method of singularities, and each of these methods ...

**24**

votes

**1**answer

1k views

### Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...

**5**

votes

**0**answers

122 views

### In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal?

Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category, all the better). ...

**5**

votes

**1**answer

668 views

### Differentiable manifolds by Serge Lang question

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...

**2**

votes

**1**answer

164 views

### Vector fields on a simplicial manifold.

Is there a known definition of vector fields on a simplicial manifold?
For me, it seems natural that the definition should be something along the lines: Let $M_{\bullet}$ be a simplicial manifold ...

**8**

votes

**2**answers

603 views

### Waldhausen $K$-theory for $G$-spaces

I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it?
Let $G$ be a finite group. We know that ...

**10**

votes

**2**answers

518 views

### When does an even-dimensional manifold fiber over an odd-dimensional manifold?

Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)?
For ...

**14**

votes

**3**answers

800 views

### Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ such that ...

**1**

vote

**0**answers

160 views

### Uniqueness of the Smooth Structure on a Handle Attachment [closed]

I posted this question on math stack exchange and didn't receive an answer. If it is too elementary for this forum I will be happy to delete it.
Let $M^m$ be a smooth manifold with boundary. We may ...

**22**

votes

**4**answers

1k views

### Is the space of diffeomorphisms homotopy equivalent to a CW-complex?

Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ ...

**9**

votes

**0**answers

244 views

### Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if
$$ du \circ j = J \circ ...

**1**

vote

**1**answer

165 views

### Can elements of Weil algebras be detected by maps into truncated symmetric algebras?

Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R.
Such algebras form the basis of the Weil approach to differential geometry, pioneered ...

**2**

votes

**0**answers

79 views

### Non-clean fiber products

Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g.
When do fibre products of smooth manifolds ...

**3**

votes

**2**answers

320 views

### if $S \times \Re$ is diffeomorphic to $T \times \Re$ then are S and T diffeomorphic?

Suppose that $S$ and $T$ are two smooth manifolds and '$ \Re$' be the reals with the normal manifold structure. And here I use '$=$' to mean diffeomorphism.
Is the statement below true?
$ S \times ...

**1**

vote

**0**answers

110 views

### Functional Analysis Generalizations: indeterminated inner product and functions over manifolds

There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?

**3**

votes

**1**answer

311 views

### Boundaries of smooth manifolds

If one has a smooth simply connected manifold $M^n$ which we know to bound an $n+1$ dimensional manifold $N$, what can be said about a handle decomposition for one in terms of a handle decomposition ...

**11**

votes

**1**answer

491 views

### Pullbacks as manifolds versus ones as topological spaces

My question is: Does the forgetful functor F:(Mfd) $\to$ (Top) preserve pullbacks?
Detailed explanation is following.
A pullback is defined as a manifold/topological space satisfying a universal ...

**1**

vote

**1**answer

142 views

### finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation
$det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$
you can find the definition of effective forms here

**11**

votes

**3**answers

710 views

### Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are ...

**1**

vote

**2**answers

370 views

### Is this surface diffeomorphic to a sphere(S^2)? [closed]

let $f:R^3 -> R \ \ \ \ \ \ \ \ f(x,y,z)=x^4 + y^6 +z^8 \\$
$M = f^{-1}(1)$
Is M is diffeomorphic to a sphere $S^2$ ?
I tried to solve this problem, but I realized that I have no tools to ...

**12**

votes

**1**answer

320 views

### What is the status of the PL-pseudoisotopy stability theorem?

Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M ...

**0**

votes

**1**answer

472 views

### fundamental group of a compact manifold

why fundamental group of of compact manifold is finitely presented

**0**

votes

**1**answer

255 views

### A $C^\infty$-function on a submanifold which is not the restriction of a a $C^\infty$ on $M$

I am looking for an example showingthat
a function $f$ which is $C^\infty$ on a submanifold $N$ of $M$, but it cannot be written as
the restriction of a $C^\infty$-function on $M$.

**0**

votes

**1**answer

242 views

### $q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X)$ ?

Definition: Let $(V,\Omega)$ be a symplectic vector space, we define
$\perp:\Lambda ^k(V^*)\to\Lambda ^{k-2}(V^{\ast})$
by $\perp(\omega)=i_{X_{\Omega}}(\omega)$
here if ...

**4**

votes

**1**answer

537 views

### Extended integral in Spivak’s Calculus on Manifolds

On page 48 of Calculus on Manifolds Spivak defines (Riemann) integration over rectangles $[a_{1},b_{1}]\times\cdots\times[a_{n},b_{n}]\subset\mathbb{R}^{n}$. Then on page 55 he extends this integral ...

**3**

votes

**3**answers

321 views

### Is the set of all smoothed closed simple curves on $\mathbb{R}^2$ a manifold?

In the studies of active contours they describe the set of all simple smooth closed curves on $\mathbb{R}^2$ to be a Riemannian Manifold $M$. The tangent space at a curve $c$, $T_cM$ is a set of ...

**6**

votes

**1**answer

347 views

### Is a smooth cubic threefold diffeomorphic to a rational threefold?

A theorem of Clemmens and Griffiths states that a smooth hypesurface in $\mathbb CP^4$ of degree three is not rational. I would like to know if nevertheless it is diffeomorphic (as a smooth real ...

**3**

votes

**2**answers

242 views

### Real analytic submanifolds of $\mathbb{R}^{n}$

Hallo,
Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...

**5**

votes

**1**answer

206 views

### Index theorems and orientability

Given a Dirac operator $D$ acting on some Clifford bundle $\mathcal{E}$ over a compact, even-dimensional, oriented manifold $M$, the Atiyah-Singer index theorem states that its index is given by ...

**2**

votes

**0**answers

212 views

### What are the current possibilities for infinite-dimensional manifolds? [closed]

According to wikipedia, by a theorem of Henderson '69, infinite-dimensional Frechet Manifolds embed as open subspaces of Hilbert Space. They need to be seperable & metric. They are generalisations ...

**22**

votes

**2**answers

893 views

### Cobordism of orbifolds?

Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...

**-1**

votes

**2**answers

400 views

### Are exotic spheres still exotic in generalised smooth spaces? [closed]

This is really more of a philosophical question, and the title is somewhat rhetorical:
Exotic spheres are a feature of smooth manifold theory, where certain spheres can have more than one ...

**1**

vote

**2**answers

202 views

### are immersions/submersions captured in generalised smooth spaces by some universal property?

Immersions & sumersions are important in differential manifolds. They rely on their definition of the construction of the tangent bundle.
I realise that generalised smooth spaces do not have a ...

**9**

votes

**2**answers

451 views

### An invariant method of stationary phase

The method of stationary phase is very well-known and employed in many areas of physics and mathematics, and, of course, included in various versions as theorem in textbooks, especially on pseudors ...

**4**

votes

**1**answer

369 views

### What manifolds can have a (non-piecewise) linear structure?

By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert ...

**3**

votes

**2**answers

455 views

### Rotation in Hyperkähler manifolds

Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...

**4**

votes

**2**answers

377 views

### Do transvers foliations induce complex structure?

Hallo,
I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...

**3**

votes

**1**answer

274 views

### Holonomy of a Kähler manifold

Hi,
I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla ...

**1**

vote

**0**answers

185 views

### HyperKaehler manifolds are Ricci-flat

Hi,
I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = ...

**1**

vote

**1**answer

281 views

### Holonomy group of a non-compact Kaehler manifold

Hallo,
I have the following question: Let $(M,I,\omega)$ be a not necessary compact Kaehler manifold of complex dimension $n$. Assume that there exists a nowhere vanishing holomorphic $(n,0)$-form ...

**0**

votes

**1**answer

428 views

### Geodesics on a twisted torus

This is a repost of a question I posted at MSE.
Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus:
There are five clear-cut ...

**16**

votes

**1**answer

499 views

### Is there a notion of a chain complex with corners?

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes.
On the topological side, although in the PL ...

**2**

votes

**2**answers

338 views

### Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold

Hallo,
I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, ...

**2**

votes

**1**answer

119 views

### Can stabilizer groups in an orbifold have global twisting?

Can stabilizer groups in an orbifold have global twisting?
For example, consider the two groups $\mathbb Z/3\times\mathbb Z$ and $\mathbb Z/3\rtimes\mathbb Z$ (where $\mathbb ...

**0**

votes

**1**answer

222 views

### Unique symplectic form in an adapted complex structure

Hallo,
I ave the following question: Due to Stenzel, Lempert, Szöke ect. we know that a Riemannian manifold $(M,g)$ admits a complex structure on an neighbourhood of the cotangent bundle. This ...

**1**

vote

**1**answer

260 views

### How can I picture antisymmetry of the Lie derivative?

It's obvious that the Lie derivative defined in terms of Lie brackets is anti-symmetric. But what is an intuitive way to visualize the anti-symmetry in the 'differentiating along a flow' definition?
...

**4**

votes

**0**answers

202 views

### Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings,
Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...

**2**

votes

**1**answer

407 views

### Definition of Sobolev spaces as a space of sections of certain type

I want to define Sobolev spaces for sections on a vector bundle, basically I want that a section will belong to the Sobolev space $W^{k,p}$ if its coordinates in any aceptable patch belong to the ...