general topology – Existence of locally finite fundamental domain with compact closure

I read this answer https://mathoverflow.net/a/251680 in MO (because I need a result like that for a technical lemma in my thesis) and I was wondering if the fundamental domain constructed there is compact or not or, at least, if we can assure that the set
$${gin G : g(overline{D})capoverline{D}neqemptyset}$$
is finite (with the notation and definitions used in the cited answer). Or, even weaker, can we assure that in the case that the topological space is just an open set of $mathbb{C}^{n}$?

I would be very grateful if anyone could answer this question or give any reference where I could find some result about the existence of locally finite fundamental domains with compact closure. Thank you in advance.

gt.geometric topology – Noncompact 3-manifold with fundamental group isomorphic to a surface group

Let $M$ be an open 3-manifold such that $pi_1 (M)$ is isomorphic to the fundamental group of a closed surface $S$. Furthermore suppose that $tilde{M} = mathbb{R}^3$, is it true that $M = S times mathbb{R}$?

Without making any assumptions about $tilde{M}$, there are counterexamples to the question, (see for instance… ), but in those cases the universal cover is not very nice.

general topology – Fundamental group of that using Seifert-van Kampen

I have this exercise:
Compute the fundamental group $pi_1(X)$ of the space $X=S^2 cup { (x,0,0) : xin (-1,1)
} cup {(0,y,0):yin (-1,1) } cup {(0,0,z):zin (1,1) }$
.

I tried that via Seifert-Van Kampen:

Let $I_x = { (x,0,0) : xin (-1,1)
}$
, $I_y = {(0,y,0):yin (-1,1) }$, $I_z = {(0,0,z):zin (1,1) }$.

Then $pi_1(X) = pi_1(S^2 cup I_x cup I_y cup I_z) = pi_1((S^2 cup I_x) cup (I_y cup I_z)) cong pi_1(S^2 cup I_x) ast_{pi_1((S^2 cup I_x) cap (I_y cup I_z))} pi_1(I_y cup I_z) cong (pi_1(S^2) ast_{pi_1(S^2cap I_x)} pi_1(I_x)) ast_{pi_1((S^2 cup I_x) cap (I_y cup I_z))} (pi_1(I_y) ast_{I_ycap I_z} pi_1(I_z)) cong ({0 } ast_{pi_1(S^0)} { 0 }) ast_{pi_1((0,0,0))} ({0 } ast_{pi_1((0,0,0))} { 0 })=({0} ast_{pi_1(S^0)} {0}) ast ({0} ast {0})$.

I know it looks ridiculous.

I don’t know how to compute this product with amalgamation.

algebraic topology – Show element of fundamental group is nontrivial

I’m learning cohomology, and I’d like to show the following:

Let $i:mathbb RP^1tomathbb RP^n$ be the usual embedding taking $(x_0,x_1)mapsto(x_0,x_1,0,dots,0)$, where $nge2$. Further, let $v:S^1tomathbb RP^1$ be the fibration $(x_0,x_1)mapsto(x_0,x_1)$. Show that $(icirc v)$ is a nontrivial element of $pi_1(mathbb RP^n,*)$.

I tried proof by contradiction: Suppose $icirc v$ is homotopic to the constant map $c$ which sends everything to $*$. Then these induce the same maps in cohomology: $v^*circ i^*=(icirc v)^*=c^*$. Recall that $H^q(mathbb RP^m;mathbb Z/2mathbb Z)=mathbb Z/2mathbb Z$ for all $0le qle m$. Let $Omega_nin H^1(mathbb RP^n;mathbb Z/2mathbb Z)$ be the nonzero element, and similarly define $Omega_1in H^1(mathbb RP^1;mathbb Z/2mathbb Z)$. Then I already know that $i^*(Omega_n)=Omega_1$. The Gysin sequence shows that $v^*(Omega_1)=0$.

I wanted to show that $c^*(Omega_1)$ is nonzero. But I feel like I don’t quite understand cohomology groups/rings enough. If we write $Omega_n=text{cls}~omega_n$, where $omega_nin Z^1(mathbb RP^n;2)$ is a homomorphism $C_1(mathbb RP^n)tomathbb Z/2mathbb Z$, then the goal is to show that $$omega_nc_#:C_1(S^1)to C_1(mathbb RP^n)tomathbb Z/2mathbb Z$$ is not a coboundary, but I haven’t been able to do this. After all, isn’t $omega_nc_#$ just the zero map?

This is Exercise 12.24 in Rotman, and includes the hint that $pi_1(mathbb RP^n,*)congmathbb Z/2mathbb Z$ and $mathbb RP^1approx S^1$. But I didn’t use either one.

at.algebraic topology – Commutator length of the fundamental group of some grope

A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra
$L_0 to L_1 to L_2 to cdots$
obtained as follows. Take $L_0$ as some $S_g$, an oriented compact surface of positive
genus $g$ from which an open disk has been deleted. To form $L_{n+1}$ from $L_n$, for each
loop $a$ in $L_n$ that generates the first homology group $H_1(L_n)$, attach to $L_n$ some $S_{g_a}$ by identifying
the boundary of $S_{g_a}$ with the loop $a$. Since the fundamental group of $S_g$ punctured
is a free group on $2g$ generators, this procedure embeds each $pi_1(L_n)$, and thus
each finitely generated subgroup of $pi_1(L)$, as a subgroup of a free group, with each
generator $a$ of $pi_1(L_n)$ becoming a product of $g_a$ commutators in $pi_1(L_{n+1})$. Hence
$pi_1(L)$ is a countable, perfect, locally free group. See What is a grope? for some motivation.

Here we build a grope $A$ in stages. At each stage put in disks with 3 handles on all the various handle curves at that stage. In other words, grope $A$ is built using disks with 3 handles uniformly throughout.

Question. Is it possible that there is a generating set $G$ of $pi_1(A)$ such
that each element of $G$ has commutator length bounded by 2?

support – How do I setup the fundamental architecture for field maintenance contracts with different number of service checkups

I’m looking for any assistance/help/advice in designing a simple application and database (Perhaps I should rather ask this question on a DBA Stack?). I need to create a simple application for a company that still makes use of a very basic Excel sheet to keep track of service and maintenance contracts, and when the next date is to perform preventative maintenance. As the sheet grows, it becomes unreadable, and very time consuming to pick out the elevators that require maintenance the coming month. The sheet doesn’t even make use of Excel functions etc. Everything is done by hand.

The company provides elevators. A client can have multiple elevators. Each elevator can have a different maintenance contract. For example elevator A requires two service check-ups per year, while elevator B only requires one check-up a year. A contract has a duration of one year, and can start any time of the year. After a year the contract automatically renews if the client doesn’t request to stop it.

Maintenance needs to be completed two months before the end of the contract. Example; Client John Doe signs a contract for two checks per contract year. The contract starts May 2021. May 2022 the next contract period starts. So the last maintenance check-up for the first period should be finished in the month of March 2022, and six months prior the first check-up should be done (September 2021). It is very important to keep track of the contract start and end dates because of invoicing etc.

Currently I’m thinking of a setup where we have a table contracts. This table contains the client_id (the one who pays). The user_id (the actual user of the lift). The elevator_id (details about the elevator), and a started_at date.

I made a quick draft with MS Access just for a graphical idea. The idea would then be to provide a date and the number of maintenance check-ups there should be in a form when creating a new contract and then automatically calculate the actual dates and create maintenance_visits rows for that contract (perhaps calling the table work_orders instead). When we near the end of the contract date and some of the work_orders' execution_date is still empty it would trigger some alerts. Same as when maintenance is coming up soon, looking at the planned_date field.

db design

I have a bit of programming knowledge but not that much experience. I hobby most of the time in my free time. I am the field engineer that have to do the elevator maintenance, meaning I am the one who use that Excel sheet the most, and it starts to annoy me a lot to try and figure out new routes for the coming week. So I thought of build something simple with Laravel so that I have a system that does the work for me and not me looking through a bunch of crazy colour coded Excel rows. The company is very small, they don’t want to buy already existing software that can help with this tasks. It is not feasible to invest in something that huge.

I hope there is somebody who would like to assist and show some samples or point out stuff that might be better if I do it completely different. Any advice and help to point me into the right direction will be very much appreciated!

at.algebraic topology – Conclusion of Hurewicz for $H_3$ without vanishing fundamental group?

Fix a space $X$, which I want to assume is a manifold. Under the assumption of simple-connectivity, Hurewicz’s theorem tells us that
$$
pi_3(X)to H_3(X,mathbb{Z})qquad (*)
$$

is surjective, hence that every homology class is represented by a map $S^3to X$. There is lots of hard work gone into worrying when homology classes are represented by embedded submanifolds, but that is not what I’m interested in here. What I want to know is:

  • Are there are nontrivial examples of manifolds $X$ with $(*)$ surjective, but with infinite $pi_1(X)$?

I don’t need the case when $pi_1$ is finite, since I have a different proof that doesn’t need this more subtle property.

An equivalent question is asking whether $H_3(tilde{X},mathbb{Z})to H_3(X,mathbb{Z})$ is surjective, for $tilde{X}$ the universal covering space. Since $pi_1(X)$ is nontrivial, then questions about the Eilenberg–Moore spectral sequence become more subtle, but I’m only looking at such a low-dimensional group that maybe things are not so bad (I don’t really understand the EM spectral sequence, even relative to my background knowledge of spectral sequences, so I don’t quite know how to start extracting information from that).

lie algebras – Order of the fundamental Group of a root system and the determinant of the Cartan Matrix

Let $mathfrak{g}$ be a finite dimensional simple Lie algebra over $mathbb{C}$. Let for some fixed cartan subalgebra $Phi$ be the root system of $mathfrak{g}$. Let $Delta={alpha_1,dots,alpha_{ell}}$ be a simple system for $Phi$. Let $P={lambdainmathfrak{h}^*mid (lambda,alpha^vee)inmathbb{Z}quad forall alphainDelta}$ be the weight lattice and $Q=Span_{mathbb{Z}}Delta$ be the root lattice. Then $Qsubseteq P$. If $w_1,w_2,dots,w_{ell}$ are the fundamental dominant weights, then $P$ is a lattice with basis $w_1,w_2,dots,w_{ell}$. Let $A$be the Cartan matrix of $Phi$.

My Question is: How to prove that the order of the group $P/Q$ is same as $det(A)$?

I could express $alpha_i$‘s in terms of $w_i$s and coefficient are Cartan integers. By inverting the system I showed that $det(A)w_iin Q$ for all $i$. But don’t know how to prove that the order is $det(A)$.

For classical cases we have a very explicit description of $w_i$ s in therm of $alpha_i$ s, there it’s not hard to see that order is indeed $det(A)$. But I’m trying to find a case free solution.

For general case I tried to use the stacked basis theorem for finitely generated ( at least free suffices) abelian groups but could not come up with and ans.

Any suggestion is welcomed and appreciated. Thank you in advance.

Questions regarding the fundamental theorem of galois theory

From what I have read, the fundamental theorem of Galois theory states that there is a bijection between subfields of a splitting field of a polynomial and subgroups of the Galois group.

One question I have is that why do the automorphisms in the Galois group have to fix the polynomial of the splitting field?

In the case where $B = A(alpha)$ where $alpha$ is the root to some polynomial f(x), and the galois group is $Aut(B/A)$, Is it because transforming the polynomial means A isn’t fixed?

Another way to ask this is that do all possible automorphisms of $B$ that fixes $A$ also fixes $f(x)$? If not, then why are those automorphisms not in the galois group?

Different definition of Fundamental Theorem of Finitely Generated Abelian Groups

I was reading about abelian groups but confused in Fundamental Theorem of Finitely Generated Abelian Groups, so I searched
some sources and become even more confused due to the different types of definition found.

1)
(from Abstract algebra Dummit and Foote)

Theorem. (Fundamental Theorem of Finitely Generated Abelian Groups) Let $G$ be a finitely generated abelian group. Then

(1) $G cong mathbb{Z}^r times Z_{n_1} times Z_{n_2} times … times Z_{n_s}$, for some integers $r$, $n_1$, $n_2$, … , $n_s$ satisfying the following conditions: (a) $r ge 0$ and $n_j ge 2$ for all $j$, and (b) $n_{i+1} mid n_i$ for $1 le i le s-1$

(2) the expression in (1) is unique: if $Gcong mathbb{Z}^t times Z_{m_1} times Z_{m_2} times … times Z_{m_u}$, where $t$ and $m_1$, $m_2$, … , $m_u$ satisfy (a) and (b) (i.e., $t ge 0$, $m_j ge 2$ for all $j$ and $m_{i+1} mid m_i$ for $1 le i le u-1$), then $t = r$, $u = s$ and $m_i = n_i$ for all $i$.

$:::2.$

(I can see that second definition is not for isomorphism but still,)
$$A = mathbb{Z}^n oplus mathbb{Z}_{h_1} oplus … oplus mathbb{Z}_{h_n}$$
$$where ::h_i | h_{i+1}$$

$: ; :3.$

To use the theorem you must factor suppose $vert G vert=n$ and factor $n$ into prime factors. Then you must get divisors $d_1,d_2,dots,d_m$ such that $d_imid d_{i+1}$, and $d_1cdotdotscdot d_{m}=n$, and your group can be

$$Gcong mathbb{Z}_{d_1}oplus mathbb{Z}_{d_2}oplus dots oplus mathbb{Z}_{d_m} $$

I looked in Wikipedia but they only say about uniqueness but don’t have these definitions with divisibility.

Can someone please explain why there is difference in the divisibility conditions in these definition and what each means.

I am new to these so please forgive my negligence.