I'm learning some algebraic topology books which introduce me to singular cubial homology and cohomology ( not simplicial as usual so sometimes it's hard for me . I think the difference between these theory is triangulation ) . I will go straight to main problem , if you have studied Poicare duality theorem you may know the way to define a orientable manifold through group homology . In higher dimension of manifold , it can't be imagined . My question is the intuition of homology and cohomology group in higher dimension . How these groups affect to orientation or somethings like this . For instance , rank of $H_{0}$ is the number of component , $H_{1}$ is abelianization of fundamental group . I try to find some ideas on Internet but the only I get is like " the number of $n$ dimension holes the space has " .  I can't happily calculate when I don't really understand what they actually mean .
I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept relatively simple, but I imagine it is entirely possible there is no real answer to my query anyway.   As I said above, I want to gain a little deeper understanding of what the n-th homology group actually means: I can happily calculate away using Mayer-Vietoris but it doesn't really give me a great deal of intuition about what the n-th homology group actually means. For example, with homotopy groups, the fundamental group is in some sense a description of how loops behave on the object in question, and it is obvious to me why that is what it is for say, the torus or the circle. However, I have no idea what, if anything, I am actually saying about a triangulable object when I talk about it having 0-th homology group this or 1st homology group that.  The best I have been able to find online or in my limited book selection is the brief description "intuitively, the zeroth homology group counts how many disjoint pieces make up the shape and gives that many copies of $\Bbb Z$, while the other homology groups count different types of holes". What 'different types of holes' are there, roughly speaking? I appreciate that it may often be completely non-obvious what the low-order homology groups are for some complicated construction, but perhaps in simpler examples it might be more explicable. Are there (simple) cases where I could say, just from looking something like e.g. the torus, what its zero-th or first or second etc. homology group was based on the nature of the object? I guess in the zero-th case it is, as my source () above says, related to the number of disjoint pieces. Can we delve deeper than this for the other homology groups?   Any book/website suggestions would be welcomed (preferably websites as I am nowhere near a library!) - I have Hatcher but not a great deal else, and I haven't gleaned as much as I wish to from that alone. Of course I know that there is a great deal we don't know about homology groups even today, so I don't expect some magical all-encompassing answer, but any thoughts you could provide would be appreciated. I hope this question is appropriate for SE Mathematics, apologies if not! -M
Starters on Stack Overflow tend to lose interest as soon as their first question starts getting downvotes. Their first question also sometimes gets closed because it was either not framed properly or it missed some very common things.  There should be way to cope with this for instance give some starter bonuses to new users. Ownership of content in Community Wiki posts How do threads work in Python, and what are common Python-threading specific pitfalls? I've been trying to wrap my head around how threads work in Python, and it's hard to find good information on how they operate. I may just be missing a link or something, but it seems like the official documentation isn't very thorough on the subject, and I haven't been able to find a good write-up.  From what I can tell, only one thread can be running at once, and the active thread switches every 10 instructions or so?  Where is there a good explanation, or can you provide one? It would also be very nice to be aware of common problems that you run into while using threads with Python. Fatou's lemma and measurable sets I don't know how can I imply Fatou's lemma for any measurable sets $A_k$, that is, $$\lambda(\liminf A_k)\le\liminf\lambda(A_k).$$ How can I prove it? And is there any example in $\mathbb R$ of sequence of measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim\lambda(A_k)=1$, but $\liminf A_k=\varnothing$ ? Thanks for your help! Can jQuery .load append instead of replace? Why is the edit button disabled? I edited some questions before, but now I get an error message when clicking the edit button. I thought there was a problem with the specific question, but I cannot click it on any question.  What's going on? Is there a dominant gene for right-handedness? Has there been any definitive research about handedness being genetic? Also, why is right-handedness clearly dominant in humans? I'm interested in evolutionary theories, as well as any molecular explanation (if known). Let $f: B(0,1) \rightarrow \mathbb{C}$ be holomorphic and suppose $\exists\ r \in (0,1)$ such that $f$ is injective in $A = \{z \in \mathbb{C} : r &lt; |z| &lt; 1\}$. Prove that $f$ is injective.  I tried using Rouché theorem, or the identity theorem, but I don't know what to do. Any hints? :) Is changing your passport a good strategy for removing a negative immigration history in Schengen? When certain nationalities of people want to travel in the Schengen zone, they are required to apply for a Schengen visa. This is a straight-forward process for most, but when an applicant has had problems with removals or refusals they may be reluctant to apply because they want to avoid the complexities of serial visa refusals. This reluctance may lead some into wondering how their history can be expunged, and this line of thought inevitably leads to the idea of getting their passport replaced with a new one.  This sounds like a great formula for people who have had prior refusals or removals.  In essence they can become 'reincarnated' as a first-time traveller and hence avoid the unpleasantness of revealing their prior interactions with the Schengen authorities.  The relevant parts of the application form are shown here...    Nothing asks about a prior removal or refusal, and the applicant is left with simple denials for 3 questions in order to present themselves as a first-time traveller.  Assuming the person is happy to set aside the moral issues associated with denying their history, are there any holes to this strategy?  This question is scoped to Schengen visa applications so that it will not be unmanageably broad, but answers from other regimes (particularly the affluent Commonwealth and the USA) are acceptable as long as the main points are relatively consistent with Schengen. Why the Principle of Least Action? What's the difference between these two sentences? And which one's grammatically correct? Why do objects in front and back view look reversed? Whenever I model objects that has different look at front and back. Viewing it from front perspective or orthographic the model displays back of the object. Parsing XML, JSON, and newer data file formats in UNIX using command line utilities The Unix environment has some excellent tools for parsing text in various forms. However, of late, the data is not in the traditional (historical) formats (CSV, TSV, record-based or some other delimiter-based) it used to be before. Data these days is exchanged in structured formats like XML/JSON.  I know there are some good tools like sed, awk and Perl which can chew down nearly any form of data out there. However, to work with this sort of structured data, often one has to write a complete program, and, given the little time available to extract information, one has to sit down and figure out the whole logic of what one wants to query and put it down programmatically. Sometimes this is not OK – basically because the information extracted from those files acts as inputs for further work; also because of the time it takes to search for the appropriate solution and code it up. A command line tool is needed with sufficient switches to find, query and dump data.  I'm looking for tools that take a XML/JSON or other forms of structured data and dump it into other formats like csv, etc., so that from there one could use other commands to get any information out of it.  Are there any command line utilities you know of which do this kind of a job? Are there already awk/Perl scripts available to this? does not appear to actually define what a rigid shape is. Rather it defines 'same shape' and 'rigid transformations' without giving any definitions of what is necessary and sufficient for a shape to be considered rigid.  For instance, I've seen the following image:    I understand intuitively why the triangle is rigid and quadrilateral is non-rigid. It is also my understanding that inserting a single diagonal connection into the quadrilateral would make the shape rigid? However, given the following image (labeling the shapes $S_1$, $S_2$ and $S_3$ in sequence left to right):    If $S_1$ had a single diagonal connection AB, it would be possible to flip ADB over axis q to obtain a shape similar to $S_2$ with the addition of connection AB (my explanation may not be the greatest). This shape is clearly not the same as $S_1$, so how can we say $S_1$ with a single diagonal connection CD is rigid? I feel I am unclear on what the definition of rigidity actually is. Distributing distinct apples and pears to three people so that each person receives at least one pear How many ways are there to distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear.   Note: The first part of the problem was the same but the word distinct was replaced with identical apples and pears. That answer is $360$ ways.   What has to be done differently when the word identical is replaced with distinct? Groups of order $56$ Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove?  My idea: consider the case $n_2=7,n_7=8$. Then the number of elements in all Sylow $2$ -subgroups and Sylow $7$-subgroups is at most exactly $(7-1)*8+(2-1)*7+1=56$. This cannot give a contradiction. We need to prove that $G$ is simple. Then $|G||(8!/2)$. To prove $G$ simple, we need to prove there does not exist normal subgroups of order $2$ and order $4$. But I do not know how to prove. I'm developing some scripts. One for the  that popped in Ubuntu 16.04 even though pcspkr is blacklisted. Another to fix a  where sound switches off from TV to the laptop.  What is the best directory to put my bash scripts in?  I've searched the Q&amp;A database but I get a bunch of different answers for a bunch of different scenarios.  /usr/local has been recommended by one person because it's in the PATH$. Is that the right thing to do?  Thank you for your advice.    Even though I accepted below answer over a year ago, I ended up using:   /usr/local/bin for scripts needing sudo powers /home/me/bin for scripts running at user level   After creating directory /home/me/bin you need to log out and in again for it to get into your PATH. What are the prerequisites for studying mathematical logic? I am looking to study mathematical logic, however, I find that introductory books are very daunting, which kind of disheartens me. You see, slowly but surely, I started to realize that the maths which I have learned did not just pop out of thin air, but is a collection of systems, which must of been developed via some other system, i.e, maths did not develop itself.  So I began to look into the origins of mathematics, and read that it was developed via a type of logic, which exists sort of by 'default', via a set of axioms, and then of course I looked up the definition of axioms.   So given that I'd be studying a type of logic whose origins are self evident axioms, naturally I believed there would be no prerequisites. However, in looking up mathematical logic, I have come across things such as Boolean algebra, sets, first order logic, some other type of logic, called 'traditional logic', as well as references to a sort of calculus, though not in a mathematical sense, I think.  So all in all, I am trying to develop a type of mental spider web, and I am trying to find out the strands which lye at the absolute bounds so that I may learn this mystique logic. Though I have no idea where to start.  Side note: This is the book I have started reading:   Credit goes to Wolfgang Rautenberg. Batch Rename (remove specific character) Files I was asked to prove the following:  If $\{ E_1,E_2,\ldots,E_n\}$ is a fully independent set of events, prove that:  $$ P\left(\bigcup_{i = 1}^n E_i \right) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right]  $$  (Hint: use complements)  I got the following:  Let $\alpha = E_1\cup E_2 \cup \cdots \cup E_n = \{E^C_1 \cap E^C_2 \cap \cdots\cap E_n^C\}^C$. Then, $\alpha^C = \{ E^C_1 \cap E^C_2 \cap \cdots \cap E^C\} $.  Assuming that the set of complements of each ${E_i}$ is also fully independent, we have:  \begin{align} &amp; P(\alpha^C) = \prod_{i = 1}^n P(E_i^C)  = \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ &amp; P(\alpha) = 1 - P(\alpha^C) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ &amp; P(\bigcup_{i = 1}^n E_i) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \end{align}  However, even though I know (by Googling it) that my assumption is true, I'm unable to prove it so I could validly use it as a lemma. Is there another way to prove this equality?