at.algebraic topology – What is the current state of research in Chern-Simons theory?

I’m a PhD student in mathematical physics looking for some orientation. As asked in the title, I would like to know the current state of research in Chern-Simons theory. More specifically, what are some of the directions that people are currently pursuing in this field.

I did not ask for more general topological field theories, but this is largely due to personal interests. Of course, answers related to other TFTs are more than welcome.

I feel this question has been asked before. But since this is a question about the current state of research, I think it deserves an update.

Thank you very much.

gt.geometric topology – “Basic” loops on standardly embedded surfaces

Take a genus $g$ surface $S$ standardly embedded in $mathbb{R}^3$, by which I mean it is unknotted. Surface $S$ bounds a volume $V$ that deformation retracts on a standardly embedded planar graph $G$ with $beta_1 = g$, and that only has degree $3$ vertices.

Among the loops on $S$ that are null homotopic in $V$, there is a subset that are boundaries of embedded disks in $V$ that intersect $G$ exactly once for some choice of $G$ as above.

Do these loops (or perhaps close variants) have a name? Do they have an alternate definition?

at.algebraic topology – Set of all sections of a fiber bundle up to homotopy equivalence

Let $pi: E to B$ be a fiber bundle of (topological or differentiable) manifolds. Denote by $(B, E)_{pi}$ the set of all homotopy classes of sections of the bundle, i.e

(B, E)_pi &= {sigma: B to E | pisigma = text{id}_B }/sim \
sigma sim sigma’ &iff exists H: I times B to E | H_0 = sigma, H_1 = sigma’, pi H_t = text{id}_B

Is it known how to calculate such set? With “calculate” I mean to reduce the computation of it to the computation of something more known, as the homology/cohomology/homotopy groups of $E$, $B$ or some combination of them.

at.algebraic topology – A generalization of integral Poincaré duality

In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $mathbb{k}$:

An augmented differential graded algebra $R$ over $mathbb{k}$ is Gorenstein if $text{Ext}_R(mathbb{k},R)$ is concentrated in a single degree and has $mathbb{k}$-dimension one.

$X$ is Gorenstein over $mathbb{k}$ if the cochain algebra
$C^*(X,mathbb{k})$ is Gorenstein.

This definition is motivated by their subsequent results on this being a generalization of the notion of a Poincaré duality space.

Does there exist a parallel notion of a Gorenstein space over $mathbb{Z}$ which similarly generalizes Poincaré duality over $mathbb{Z}$?

general topology – How to show that a (metric) space having a countable dense subset is a topological property?

I have to show that a (metric) space having a countable dense subset is a topological property.

Given that A property P of a space is said to be a topological property if home-omorphic spaces share the same properties.

I think if i can show that two seperable spaces are homeomorphic then i can say that this is a topological property. I need help to understand this and how to show that?

Is R connected in topology generated by $B_1$ = {[a, b] : a, b ∈ Q}, and $B_2$ = {[a, b] : a, b ∈ R}

Consider the following collections of subsets of R:

$B_1$ = {[a, b] : a, b ∈ Q},

$B_2$ = {[a, b] : a, b ∈ R}.

Already show that they are two basis and let their topology be $T_1,T_2$.
And T⊂$T_1$$T_2$ where T is usual topology.

Asked to determine if $R$ is connected in $T_1$ and $T_2$.
Have no idea where to start, I think it is clearly connected in these topology but how to actually prove it?

gn.general topology – When is a function not a local homeomorphism?

my background is engineering and I am very new to the topology.

I can understand the concept of a local homeomorphism, but I cannot come up with a concrete example that is a continuous surjection but not a local homeomorphism.

For example, if I have a map $f:mathbb{R}^nto Usubsetmathbb{R}^m$ ($n>m$) that is a continuous surjection, can it not be a local homeomorphism?

gt.geometric topology – Books on Foliations

I am looking for resources (books, notes, lecture video, etc. anything will do although printed material in English is preferable) on foliations which satisfy some or all of the following constraints.

  • Prerequisites: I am familiar with algebraic topology (in the geometric style, as in Hatcher), differential topology (as in Guillemin-Pollack), and Riemannian geometry (as in do Carmo) along with all other standard undergraduate topics (by which I mean content covered in standard textbooks in real and complex analysis, linear and basic algebra, commutative algebra, classical algebraic geometry, point set topology, curves and surfaces). I am also familiar with characteristic classes (Morita’s differential forms book and Madsen-Tornehave), basic symplectic geometry (da Silva), and basic topological/measure theoretic dynamics (earlier chapters of Brin-Stuck).

I am also familiar with physics (general relativity using Caroll, classical mechanics using Goldstein, etc.) if it helps.

Ideally, I am looking for two types of resources. (Notice that the two are not mutually exclusive.)

  • An exposition of the theory which has a strong geometric taste (much like Hatcher’s books) ideally with a lot of pictures and concrete examples. Ideally, the book connects new ideas introduced in the book with older ideas (described in “prerequisites” above).

  • Collection of problems which allows one to practice applying the theory. I prefer exercises which are not just filling in technical details which the author did not have time for. Instead, I prefer something which allows one to a.) learn key heuristics, and ideally b.) get a sense on why the theory will be important later in one’s studies.

So far, I have the following books:

  • Tamura, Topology of Foliations: An Introduction
  • Calegari, Foliations and the Geometry of 3–Manifolds

2016 – Search Topology – too many databases and one corrupt

Our organization was having an issue with our Search Service App on our SharePoint 2016 On-Prem environment. Once we got it working properly, we noticed that there were 3 sets of topology component databases than our expected two. Then, one of our CrawlDBs became suspect. Any recommendations on how to move forward? It would make sense to me to transfer the current topology component databases to the orphan ones and then remove the corrupt, but I don’t know if this is best practice. Much appreciated.

general topology – Show that $g:Xto Z$ where $g((x,n)) = ((x,nx))$ is not Quotient.

Suppose X is a set of lines $$ L_{n} = mathbb R*text{{n}} $$ for $ninmathbb Z^{+}$ and $Z$ is a set of lines that cross the center of plane and its slope is positive. $$ L_n^{‘} = {(x,nx);ninmathbb Z^{+}}$$
Show that $g:X to Z$ $g((x,n)) = ((x,nx))$ is not Quotient map.
By definition if I show that for some subset of $Z$ like $W$, $p^{-1}(W)$ is open in $X$, but $W$ is not open in $Z$, then it is not quotient.
I don’ know how to show it.