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3
votes
1answer
331 views

Determining continuous functions on Banach spaces

Let $X$ be a real Banach space. For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
5
votes
2answers
885 views

Construction of Serre Spectral Sequence

I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me. He starts with considering a double complex $C_{\bullet,\bullet}$ with ...
12
votes
3answers
454 views

Recognizing the 4-sphere and the Adjan--Rabin theorem

The problem of recognizing the standard $S^n$ is the following: Given some simplicial complex $M$ with rational vertices representing a closed manifold, can one decide (in finite time) if $M$ is ...
5
votes
0answers
524 views

Homotopy groups of a Bouquet of n-spheres

Let $$X=\vee_{\alpha\in A} S_{\alpha}^n$$ be a bouquet of $n$-spheres. Q: How does one compute the homotopy groups $\pi_k(X)$?
-1
votes
1answer
420 views

Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy 1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $. 2) $ a\bar{b}+c\bar{d}=0 $ There is a ...
5
votes
2answers
786 views

Two Definitions of “Character” of topological groups

When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows: Let $G$ be a topological group. A character of $G$ is a ...
0
votes
1answer
312 views

A form of Lefschetz duality

Let W be a manifold with boundary such that \partial W is a union of two compact manifold A,B attached along their boundary. Does poincare duality hold for (W,A) and (W,B)?
5
votes
4answers
374 views

degenerating immersion

Hi, I would like to know if it exists a sequence of $C^2$ immersion $f_k: S^2 \rightarrow \mathbb{R}^3$ which converge (in C^2) to $z^2$ except on a finite set of point, i.e $f^k \rightarrow z^2$ in ...
1
vote
1answer
691 views

A question about density character of Banach spaces. [closed]

Let $\langle M_i:i<\theta\rangle$ be an increasing chain of Banach spaces, where each $M_i$ has density character $\mu$ (i.e.,the mininum cardinality of a dense subset of $M_i$ is $\mu$). Let ...
1
vote
3answers
477 views

Ultrafilters and principal filters [closed]

Can someone give me an example of an ultrafilter which is not principal?
3
votes
1answer
315 views

Topological space with some conditions

Can one give an example of non-compact space $X$ which satisfies the following conditions: the countable union of compact subsets is relatively compact, for every closed noncompact subset $A$ of $X$ ...
2
votes
2answers
555 views

Are presheaves of constant functions sheaves?

Hey there, I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ ...
7
votes
2answers
356 views

The topology of open semi-algebraic sets (appl.: totally positive matrices)

Let $P_1,\ldots,P_r$ be polynomials over ${\mathbb R}^N$. I am interested in the homotopy class of the semi-algebraic set defined by $$P_j(x_1,\ldots,x_N)>0,\qquad j=1,\ldots,r.$$ Is there a ...
-11
votes
1answer
763 views

Direct product of filters

Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$. I will denote the principal filter ...
3
votes
1answer
431 views

Morse theory and adiabatic limits

Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and ...
-8
votes
2answers
973 views

Filters and intersection of two binary relations

Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered inverse to set-theoretic inclusion. I will denote $\left\langle f \right\rangle \mathcal{X} ...
0
votes
1answer
292 views

Sufficient conditions for Hausdorffness

Let $(X,\tau)$ be a $T_1$ topological space and $Y\subset X$ a dense subspace which is completely metrizable. Are there any sufficient conditions to ensure that $(X,\tau)$ is Hausdorff using the known ...
2
votes
1answer
233 views

Sequences of groups, exact not just in etale but also in the Zariski topology

Given $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X=Spec(A)$. $B$ denotes a free $A$-algebra of rank $e^2$, actually we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv=\xi_e vu$, where $\xi_e$ is an e-th ...
2
votes
0answers
271 views

Bipartite Obstructions to genus $g+1$ Bipartite graph from becoming genus $g$.

Say $B_{n,n}$ is a bipartite graph on $2n$ vertices with each color assigned to $n$ vertices. Say I know $g \le genus(B_{n,n}) \le g+1$: What obstructions can $B_{n,n}$ have from becoming a genus ...
8
votes
0answers
460 views

Characterization of Unusual Topologies of $\mathbb R$

Following some argument over a question on math.SE, I began to wonder: We all know that in the standard topology there are no continuous bijections from $[0,1]$ to $(0,1)$ (for example by arguments ...
0
votes
1answer
309 views

Triviality of finite fiber bundles [closed]

Hello, I suspect the following is true and easy but I am unable to prove. Suppose (E, B, π, F) is a fiber bundle, where E,B are compact and F is finite, prove that E is a trivial fiber bundle. Any ...
0
votes
1answer
423 views

Extending Continuous Sublinear maps on dense subsets of a Banach space

Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying: (1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||. Please prove ...
21
votes
1answer
2k views

Math and Wormholes

Hopefully Math Overflow is the correct place for this. I had a student approach me ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any ...
20
votes
1answer
902 views

Square roots of $\mathbb R^{2n}$

Recently, Richard Dore asked us if $\mathbb R^3$ is the cartersian square of some space, and Tyler Lawson answered beautifully in the negative. The even powers of $\mathbb R$ were left out in that ...
2
votes
1answer
345 views

Homology of a region of the plane

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let ...
3
votes
1answer
135 views

Examples of H-cogroups and a question about julia sets

The following questions should be very easy, but I need help. Notations are same as Bredon's geometry and topology book. This question is related to chapter VII.3.page 441. $\nabla \colon X\vee ...
9
votes
4answers
818 views

When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
12
votes
5answers
1k views

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 ...
2
votes
0answers
60 views

Characterizing local homeomorphisms into an exponent

Let $X$,$Y$, and $Z$ be (compactly generated) spaces. Suppose $f:Z \to Y^X$ is a local homeomorphism. How can we tell this from its adjoint $\tilde f:Z \times X \to Y$? I.e., I want a property $P$ ...
4
votes
3answers
423 views

Flat regions on surfaces of genus greater than 1

Here is a polygonal disk + gluing scheme model of a surface we are attempting to construct. We want the regions of the surface bounded by the two vertical, dotted lines $\alpha,\beta$ to have zero ...