## gt.geometric topology – Image of the pure braid group under Milnor’s \$barmu\$-invariants?

As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms.

There are a relationship between the mapping class group and the pure braid group, which the Johnson homomorphism corresponds to the Milnor’s $$barmu$$-invariant or the Artin representation into not $$operatorname{Aut}(F)$$ but the automorphism group of free nilpotent quotient $$operatorname{Aut}(F/gamma_k(F))$$ where the lower central series $$gamma_k(F)$$ of a free group.

(It is known that the image of the pure braid group under the injective Artin presentation into $$operatorname{Aut}(F)$$.)

Then, is it also unknown that what is the image of the pure braid group under the Artin presentations or the Milnor invariants?

## gn.general topology – Necessity and sufficient condition of perfectly normality

In Engelking’s book, there is exercise (p. 49, ex. 1.5.K), where is written that T1 space X is perfectly normal if and only if for every open set W from X where exist sequence W1,W2,… of open subsets of X such that W is the union of Wi and cl(Wi) is in W for any i=1,2,…
=> implication is not hard to prove. is right proposition holds, it’s also obvious that X is perfect space (every open set is F-sigma set). But I can’t prove that X is normal also. Thank, for any help.

## gn.general topology – Stronger form of countable dense homogeneity

I am completing my undergrad thesis about topological properties of some subspaces of the real numbers, and CDH spaces are one of the topics I´ve covered (I know almost nothing about it, I only prove that the Cantor space, the set of irrational numbers and $$mathbb{R}$$ are CDH). However, I think I was able to prove a stronger version of countable dense homogeneity for the Cantor set. Specifically:

Let $$A_n$$ be a sequence of disjoint countable dense subspaces of the Cantor set $$2^omega$$, and let $$B_n$$ be other such sequence. Then there is an homeomorphism $$f:2^omegato2^omega$$ such that $$f(A_n)=B_n$$ for all $$n$$.

If this property is true for $$2^omega$$, it is not difficult to prove that it is also true for the Baire space $$omega^omega$$, and I think I was also able to find other proof that $$mathbb{R}$$ has this property too (probably manifolds will have it too but that´s outside the scope of my thesis). I was going to include this property and it would be useful to reference some paper about the topic, but I found nothing about it in a few papers I looked up about CDH spaces.

So my question is, is there any paper/book where I can find information about this property, or about any other similar properties or concepts that are easily seen to imply this one?

## Basis of Euclidean topology on \$mathbb{R}\$ such that no element is contained in another

What is an example of a topological base $${cal B}$$ for $$mathbb{R}$$ with the Euclidean topology such that for every $$B_1neq B_2 in {cal B}$$ we have $$B_1notsubseteq B_2$$?

## gt.geometric topology – Circle-valued Morse function and minimal genus

I think the following two statements are true, and most likely are in the literature. If so, could someone point me to some references? If not, counterexamples?

1. Let $$Y$$ be a closed oriented connected 3-manifold, and $$(theta)in H^1(Y;mathbb{Z})$$ be a primitive class. Then a minimal genus embedded surface representing the Poincar’e dual of $$(theta)$$ may be chosen to be connected.
2. In the preceding context, suppose $$theta=df$$ where $$f$$ is a circle-valued Morse function with no extrema. Then one of the regular level surfaces of $$f$$ is a minimal genus embedded surface representing the Poincar’e dual of $$(theta)$$.

## general topology – Winding of a series of complex terms exp(i k_n x) with incommensurate frequencies k_n

Assumptions and definition of the problem:

I consider the complex function
$$f(x) = sum_{n=0}^M a_n exp(-i k_n x),$$
where $$M>2$$ is a finite integer, $$x$$ is a real-valued number, $$k_n$$ is a set of mutually incommensurate frequencies with $$k_{n+1}>k_n$$ and $$k_0=0$$, $$a_n>0$$ is a set of real-valued weights with $$a_n and
$$sum_{n=0}^M a_n = 1.$$

Numerical evidence:

Studying the function $$f(x)$$ in the complex plane, a host of numerical examples show that the unwrapped phase diverges to $$-infty$$ for $$xrightarrow infty$$. Note that for all numerical studies I used commensurate frequencies $$k_n$$, yielding a periodic function $$f(x)$$. The figure below shows an example with a period of 30 (you find here an animation). In the example in the figure, $$|f(x)|neq 0$$ for all real-valued $$x$$. In other examples, whenever $$f(x)$$ reaches a zero, then the unwrapped phase is assumed to jump by $$pi$$.

Conjecture:

I am looking for a rigorous proof that the unwrapped phase does indeed diverge to $$-infty$$ for $$xrightarrow infty$$. In other words, I would like to prove that the function $$f(x)$$ winds an infinite number of times around the origin anti-clockwise. Note that whenever $$f(x)$$ reaches a zero, the so-called unwrapped phase is assumed to jump by $$pi$$, by definition.

Observations:

• It is important that $$a_0<1/2$$. On the contrary, if we have $$a_0>1/2$$, the normalization condition on $$a_n$$ imposes that no winding occurs. In fact, the weights $$(a_0,a_1,a_2,ldots)$$ can be continuously transformed to $$(1,0,0,ldots)$$, which corresponds to the trivial function $$f(x)=1$$, without that $$f(x)$$ ever crosses the origin.

• It is important that $$k_n$$ are mutually incommensurate. Otherwise, we can find counterexamples for which no winding occurs. An example is provided by $$k_n = n$$ and weights distributed according to a Poissonian distribution,

$$a_n=exp(-lambda)frac{lambda^n}{n!}.$$

I am interested in this problem for my research work in physics, because it is linked to the behavior observed in some experiments, where we probe the quantum wave function of an atom in a trap potential.

## at.algebraic topology – Why the symbol map in Atiyah-Singer paper is continuous?

I am reading Index of elliptic operators:I paper, by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512 :

Let X be compact and E,F are vector bundles over X, then the symbol
$$sigma:mathcal{P}_s^m(X;E,F) rightarrow text{Symb}^m(X;E,F)$$
is continuous for the sup norm topology on the unit sphere bundle of $$T^*X$$; it extends by continuity to a map $$sigma_s:overline{mathcal{P}_s^m}(X;E,F) rightarrow overline {text{Symb}^m}(X;E,F)$$ with kernel equal to compact operators $$H_s rightarrow H_{s-m}$$.

Notation. $$mathcal{P}_s^m(X;E,F)$$ is space of pseudo-differntial operators of order $$m$$ considered as a map $$H_{s}(X;E) rightarrow H_{s-m}(X;F)$$
and $$overline{mathcal{P}_s^m}(X;E,F)$$ is its completion under the operator norm topology.

I know how to prove that when $$m=s=0$$ and I know how to show that the mentioned kernel is composed of compact operators, I can show also that it contains all differential elliptic operators of order m-1 ( I wonder if the closure of the latter is exactly the compact operators).

I will be thankful for any help, It would be great if you give me a hint how to prove this when X is euclidean domain and $$E,F$$ are trivial line bundles.

## at.algebraic topology – Rational homotopy groups of \$S^2vee S^2\$

From what I understand $$pi_n(S^2vee S^2)otimesmathbb{Q}neq 0$$ for $$ngeq 2$$. My question is:

Is there a “hands-on” proof of this fact using differential forms?

I am sure I will receive answers like: that is Hilton’s theorem or use Sullivan’s minimal model or check the section in Bott and Tu about the rational homotopy theory.

However, all these answers are useless for me because I am an analyst and not topologist and in order to use this fact in my research I need a straightforward construction that I could use to get integral estimates of forms.

By explicit I mean as explicit as the degree defined as the integral of the pullback of the volume form or the Hopf invariant as the Whitehead integral formula.

## gn.general topology – Is every path connected \$F_sigma\$ subset of a plane an image of \$[0,1)\$?

No, this fails even for compact subsets of $$mathbb R^2$$. Namely, let $$X=Ctimes(0,1)cup(0,1)times{0}$$, where $$C$$ is the Cantor set. It is clearly path connected. $$X$$ cannot be an image of $$(0,1)$$, because the image of any interval $$(0,a),a<1$$ by this map can contain only finitely many points of $$Ctimes{1}$$ (because of compactness), and hence the image of $$(0,1)$$ can only contain countably many of them.

It might be of your interest that there is a complete topological classification of spaces which are images of $$(0,1)$$, namely they are the path-connected spaces which are countable unions of Hahn-Mazurkiewicz spaces (which means they are compact, Hausdorff, connected, locally connected, metrizable spaces), as shown here.

## at.algebraic topology – Available frameworks for homotopy type theory

I am thinking about trying to formalise some parts of classical unstable homotopy theory in homotopy type theory, especially the EHP and Toda fibrations, and some related work of Gray, Anick and Cohen-Moore-Neisendorfer. I am encouraged by the successful formalisation of the Blakers-Massey and Freudenthal theorems; I would expect to make extensive use of similar techniques. I would also expect to use the James construction, which I believe has also been formalised. Some version of localisation with respect to a prime will also be needed.

My question here is as follows: what is the current status of the various different libraries for working with HoTT? If possible, I would prefer Lean over Coq, and Coq over Agda. I am aware of https://github.com/HoTT/HoTT, which seems moderately active. I am not clear whether that should be regarded as superseding all other attempts to do HoTT in Coq such as https://github.com/UniMath. I am also unclear about how the state of the art in Lean or Agda compares with Coq.