# Tagged Questions

**4**

votes

**2**answers

173 views

### Question about lower homology class of cobordism

Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.
We know that for the highest homology class, ...

**-3**

votes

**2**answers

129 views

### The boundary of this set is piecewise smooth? [closed]

Consider a sequence of open sets in $R^n$: $\Omega_1 \supset \Omega_2 \supset\cdots$. Consider that this sets are bounded, convex with the boundary piecewise smooth .When i say smooth i mean ...

**1**

vote

**1**answer

216 views

### (n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
...

**1**

vote

**0**answers

82 views

### Proving that two given functionally structured spaces are isomorphic

The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...

**3**

votes

**2**answers

268 views

### Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...

**9**

votes

**0**answers

284 views

### 3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?

**2**

votes

**1**answer

297 views

### Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$
disjoint ...

**0**

votes

**1**answer

193 views

### Diffeomorphisms of a surface in terms of generators.

I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism ...

**1**

vote

**1**answer

312 views

### Homology and homotopy of a surface

Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$
My question is; does this ...

**0**

votes

**1**answer

206 views

### Relation on the set of connected components of the $\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition

Let $X$ be a smooth variety with an action of $\mathbb{C}^*.$ One has the so-called Bialynicki-Birula decomposition of $X$ given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where $N$ varies in the ...

**18**

votes

**1**answer

616 views

### Computing Self-Intersections with Complex Analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...

**3**

votes

**1**answer

328 views

### When is a Topological pushout also a Smooth pushout?

I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...

**57**

votes

**5**answers

3k views

### Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...

**5**

votes

**1**answer

248 views

### Characteristic Classes of a Fibered Sum

I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general.
Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for ...

**5**

votes

**2**answers

200 views

### Simultaneously minimizing intersections

This may be a standard problem in homotopy theory, but I don't know a good reference.
Let $\Sigma$ be a smooth, oriented surface, and let $X_1,X_2$ and $X_3$ be three smoothly embedded curves in ...

**2**

votes

**1**answer

496 views

### Question about analytic curves

Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true ...

**0**

votes

**1**answer

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] ...

**1**

vote

**2**answers

615 views

### Topological degree theory

Let $D$ be a region in $R^n$. If $f:D\to R^n$ is continuous, nonzero on $\partial D$ and of Brower degree 0, does there exists a continuous function $g=f$ on $\partial D$ and $g\neq 0$ on $D$?

**1**

vote

**1**answer

176 views

### When do maps of ineffective orbifolds descend to their effective part?

If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...

**3**

votes

**2**answers

258 views

### Discriminant locus in knot space

Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$.
The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers".
Let $f$ be a knot with $n$ double ...

**19**

votes

**3**answers

2k views

### Topology of function spaces?

Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let ...

**6**

votes

**2**answers

638 views

### Tangent bundle of the long line

Question: Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$? I'd guess that the answer doesn't depend on choice of differentiable structure, but maybe it does.
...

**6**

votes

**2**answers

848 views

### Compactification of a manifold

This is just a curiosity and the question is really foggy. I'm wondering if there can exist a notion of "minimal smooth compactification" (when I say minimal I think something like adding a finite ...

**3**

votes

**2**answers

411 views

### SU(2) representations of alternating knot groups

Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...