Let $M$ be a closed connected 3-manifold, and write $H_1(M, \mathbb{Z})$ as $\mathbb{Z^r} \oplus T$ where $T$ is torsion. Show that $H_2(M, \mathbb{Z})$ is $\mathbb{Z^r}$ if $M$ is orientable or $\mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$ if $M$ is not orientable. I tried applying the UCT for Homology but I am not getting any information about $H_2$. On the other hand, when I used UCT for cohomology I am unable to proceed because of the $Ext(*)$. Can anyone help?
I'm trying to solve this problem from Hatcher 3.3.24.  Let $M$ be a closed connected 3-manifold, and write $H_1(M, \mathbb{Z})$ as $\mathbb{Z^r} \oplus T$ where $T$ is torsion. Show that $H_2(M, \mathbb{Z})$ is $\mathbb{Z^r}$ if $M$ is orientable or $\mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$ if $M$ is not orientable. In particular $r \geq 1$ if $M$ is not orientable. Using exercise 6 construct examples showing there are no other restrictions on the homology groups of closed 3-manifolds. (In the nonorientable case consider the manifold $N$ obtained from $S^2 \times I$ by identifying $S^2 \times \{0\}$ with $S^2 \times \{1\}$ via a reflection of $S^2$.)  I think I've solved the first part, here's my proof:   First note the following theorem: If $M$ is a closed connected and orientable n-manifold then $H_n(M, \mathbb{Z}) = \mathbb{Z}$ and $Tors(H_{n-1}(M, \mathbb{Z})) = 0$. If nonorientable, then $H_n(M, \mathbb{Z}) = 0$, $Tors(H_{n-1}(M, \mathbb{Z})) = \mathbb{Z_2}$.  Now note by Poincare duality and UCT we have that $H_2(M) \cong H^1(M) \cong \mathbb{Z^r} \oplus T_0 \cong \mathbb{Z^r}$.   In the nonorientable case consider the exact sequence:  $0 \longrightarrow H_2(M) \longrightarrow H_1(M) \longrightarrow H_0(M) \longrightarrow 0$  But we know $H_1(M) = \mathbb{Z^r} \oplus Z_2$ from the earlier theorem and $H_0(M) = \mathbb{Z}$.   Therefore $H_2(M) \cong ker(\mathbb{Z^r} \oplus Z_2 \rightarrow \mathbb{Z}) \cong \mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$.   For the rest of the question I'm quite lost.
Why is ++i considered an l-value, but i++ is not? How to initialize all members of an array to the same value? To hyphenate or not? Does the Planck scale imply that spacetime is discrete? On a quantum scale the smallest unit is the , which is a discrete measure.  There several question that come to mind:   Does that mean that particles can only live in a discrete grid-like structure,  i.e. have to "magically" jump from one pocket to the next? But where are they in between? Does that even give rise to the old paradox that movement as such is impossible (e.g. )? Does the same hold true  for time (i.e. that it is discrete) - with all the ensuing paradoxes? Mathematically does it mean that you have to use difference equations instead of differential equations? (And sums instead of integrals?) From the point of view of the space metric do you have to use a discrete metric (e.g. the ) instead of good old Pythagoras?   Thank you for giving me some answers and/or references where I can turn to.  Update: I just saw this call for papers - it seems to be quite a topic after all: Is Reality Digital or Analog? FQXi Essay Contest, 2011. , , . One can find some pretty amazing papers over there. When trying to view  via the iOS app, I get this:        It's consistent, retrying does not help and the request always fail.  All other questions load just fine, so I highly suspect the cause is the Emoji icons used in there, especially in the comments that contain lots of them.  Beta channel, version 1.0.1.85  Update: still happening in 1.0.1.88 and I also found that the "preview" seen for example in tag search shows the Emoji just fine: Salesforce1 on focus scroll to the top Using \footnote in a figure's \caption Maybe this is an easy one, but I struggled with this now too long :)  I want to have a footnote in a caption of a figure, see the example.   \begin{figure}[!ht]   \caption{a figure caption\footnote{where i got it from}}   \label{somelabel}   \begin{center}        \pgfuseimage{...}   \end{center} \end{figure}   The compilation error reads as follows  ! Argument of \@caption has an extra }. &lt;inserted text&gt;              \par  l.192 ...i got it from}}   The actual tex code for my figure with the answer of Leo  \pgfdeclareimage[width=6cm]{aba.medcenter}{aba.medcenter}  \begin{figure}[!ht]   \begin{minipage}{\textwidth}     \caption[Medcenter Monthly Medication System]{Medcenter\textsuperscript\textregistered Monthly Medication System\footnote{Quelle Bild: http://www.amazon.com/dp/B000RZPL0M}}     \label{aba.medcenter}     \begin{center}       \pgfuseimage{aba.medcenter}     \end{center}   \end{minipage} \end{figure}   leads to the error  ! LaTeX Error: Command \itshape invalid in math mode.   in the same line. If I comment out the foot note, everything compiles fine. \textsuperscript\texttrademark isn't the problem, too. I'm looking for either a single word or phrase that would describe either someone or the action of dismissing someone's opinions as something "socially unpopular", without giving any reasoning why, either just to spite them, because they have a poor counter-argument, have been backed into a corner, or something similar.  For example, someone who supports something politically incorrect (like, say, stronger gun control or help for male domestic abuse) might have their views dismissed as racist or sexist or liberal without actually acknowledging and debating against them, thus labeling the former as something they might not be.  I was talking to a friend earlier about politics, and she complained about Germany's immigration problem. She said one issue was that many politicians that try to support stronger immigration laws would immediately be shot down as racist or a nazi and forced to resign, instead of being presented with counter-arguments. Regardless of whether you agree with them or not, what would describe that?  EDIT: Forgot to specify -- the word or phrase I'm looking for would be used in an informal context. And to address further questions, to be used when the person's opinion has legitimate worth and isn't just "being racist" etc, i.e. the dismisser is incorrect, and is only making these accusations because they can't argue further. (Added emphasis to the last sentence of the first paragraph) Doing bad in undergraduate but good in a masters program Suppose you do bad in undergraduate school in say computer science. But you do very well in a masters program in computer science. If you want to apply to a PhD program in computer science, will the masters degree grades offset the undergraduate degree grades? Does Linux have the capability to use "junctioned" symbolic links? I'm not sure if this is an actual term or not, so let me explain the concept.  I have a git repository containing all my configuration dot files in ~/dotfiles. I use symbolic links into this directory in order to "activate" them. For example, by executing ln -s ~/dotfiles/bash/bash_profile ~/.bash_profile to produce the link:  ~/.bash_profile -&gt; ~/dotfiles/bash/bash_profile   However, I find myself in a situation where I want to combine the contents of multiple files. For example, I want the ~/.bash_profile symbolic link to point to two separate files, one for each project. E.g.:  ~/.bash_profile (1) -&gt; ~/dotfiles/bash/bash_profile (2) -&gt; ~/dotfiles/proj/bash_profile   I know I can simply concatenate the two files (e.g., cat ~/dotfiles/{bash,proj}/bash_profile &gt; ~/.bash_profile), but if I can do the same thing with symbolic links, I would prefer to.  I imagine that if such a feature exists (Nix is pretty huge), then under the hood it would have to map the two different files together, hiding all sorts of complexities under the hood (mapping file offsets of all non-first "mapped" files, locking all files when writing to the junctioned symbolic link, etc, etc).  If such a feature doesn't exist, are there are plans to implement it? Let $C_k$ be the set obtained in the $k-$th stage of building the Cantor set, where  $$C_1=[0,\frac{1}{3}]\cup[\frac{1}{3},\frac{2}{3}]$$  $$C_2=[0,\frac{1}{9}]\cup[\frac{2}{9},\frac{1}{3}]\cup[\frac{2}{3},\frac{7}{9}]\cup[\frac{8}{9},1]$$  $$\dots$$  So we have   $$C_k=\bigcup_{j=1}^{2^k}I_{k,j}$$  where $I_{k,j}$ are pairwise disjoint closed intervals of length $3^{-k}$ in $[0,1]$, let  $$[0,1]\setminus C_k=\bigcup_{j=1}^{2^k-1}J_{k,j}$$  where $J_{k,j}$ are pairwise disjoint open intervals and each $J_{k,j}$ are located between intervals $I_{k,j},I_{k,j+1}$  define  $$F_k:[0,1]\to[0,1]$$  where  $$F_k(0)=0,F_k(1)=1,F_k(x)=\frac{j}{2^k}\quad x\in J_{k,j}$$  and $F_k$ is linear on each subinterval of $C_k$. We can show that $F_k$ converges to the Cantor function $F:[0,1]\to[0,1]$. Moreover, it can be shown that  $$|F_k(x)-F_k(y)|\leq\frac{3^k}{2^k}|x-y|$$  How can we show that $F$ is Hölder continuous of order $\log_32$ ? How can I take a screenshot with Selenium WebDriver? Why do we need to fork to create new processes? Why are doctors addressed as Mr. in the UK? In the US most physicians, surgeons and dentists are addressed as "doctor". Very few other professionals receive the same title.  In the UK, however, surgeons and dentists seem to prefer to be referred to as Mr. If that is true, why?  And who is addressed as Dr. in the UK?  Those who hold doctoral degrees only? In all the discussions about how the heavy elements in the universe are forged in the guts of stars and especially during a star's death, I usually hear that once the star begins fusing lighter atoms to produce iron (Fe) that's the end of the star's life and the whole system collapses onto itself; and based on how massive the star was initially, it has different outcome - like a white dwarf, a neutron star or a black hole. I have rarely heard a detailed explanation of how the elements heavier than iron are produced. I would appreciate a convincing explanation of this process. How should one maintain and update a MiKTeX installation? Why was "An Update to our Community and an Apology" deleted? Recently, Sara Chipps posted her message . After reading it, I wanted to compare it with . It turns out that she deleted it. Does anyone know why? It looks strange since (as far as remember) it was a sort of official position of SE. Why would such a statement be deleted? Setting up a bicycle for someone with restricted knee articulation My father, now in his 70's, had a left knee replacement several years ago. He has limited movement in that knee - probably about 85 degrees on a good day. His knee can extend fully, but cannot bend as much as it should. As a result, he cannot pedal a bike he cannot get his left leg over the top of the pedal stroke.   He is desperate to get back onto a bike, he knows he won't be ripping up the road and winning races like the old days, but wants the freedom to go for a slow easy rides.    Does anyone have suggestions as to a solution to the problem. We don't mind spending money if we know the solution will work, but he can't afford to throw big money at wild ideas that may or may not work. If we find a solution, a new bike is likely.   We have access several mountain bikes in a range of sizes to trail ideas, and we have all the tools and skills to build bikes up from parts and rearrange things as needed.    EDIT: Thanks everyone - all great suggestions. What am am going to do is got tto the LBS and get hold of a cheap set of short cranks for around $30- probably from a kids bike, in the 120-150 size range, and see if he can ride that. Depending how he goes, I'll try a standard right crank and a short left crank. If all else fails, we will go for the Power Cranks   UPDATE: In case anyone else has similar problems. Got some 152mm cranks for $30. Set an old  mountain bike up with the seat as far back as possible. The old man is able to ride reasonably comfortably, now has a smile from ear to ear - any suggestions how to remove it - doesn't fit with the grumpy old man image :)   Has got me wondering how many people with minor knee problems would benefit from short(er) cranks....... When I visit a site I haven't visited (while logged in) for a while, I see:  Welcome back! If you found this question useful, don't forget to vote both the question and the answers up.  That's often intended behavior. But today I visited  and was shown that banner even though the question and all its answers are locked. Voting arrows aren't even shown on any of the posts on that page. I believe that banner should be disabled on pages where there is no post that can be voted on by anyone. Make browser window blink in task Bar How do I make a user's browser blink/flash/highlight in the task bar using JavaScript? For example, if I make an AJAX request every 10 seconds to see if the user has any new messages on the server, I want the user to know it right away, even if he is using another application at the time.  Edit: These users do want to be distracted when a new message arrives.