@PearlSek if you want a single definition, here's one, although it's a bit funny if this was the answer that would be accepted as correct- $f(x) = \frac{|x|-x}{2x}\cdot \left(-\frac{1}{x}\right) + \frac{|x|+x}{2x}\cdot x
@BalarkaSen sorry if this is just being pedantic, but here, $F$ should go to $S^1 \times (0,1)$, right? if yes, isn't a deformation retraction usually going to a subspace?
@Huy Yes and no. If $A \subset X$, a deformation retract of $X$ to $A$ is a homotopy $F : X \times I \to X$ between the identity map so that $F(-, 1)$ is a retract (image of $F(-, 1)$ sits in $A$).
$F$ is simply a deformation retraction of $S^1 \times I$ minus center circle to $\partial{S^1 \times I} = S^1 \times \{0, 1\}$, in short. You can write down such a def. ret. by just working with the longitudes ($x \times [0, 1]$) of the cylinder.
Well, I have a specific retraction in mind (obtained by pushing the two components in the complement of center circle longitudinally towards the boundary) but sure.
"retraction of S^1 x {0, 1}" is an odd parsing though.
@BalarkaSen: sorry, I'm still a bit confused by your notation. just for the start, $F$ is the homotopy between the identity and the retraction $r: S^1 \times (0,1) - S^1 \times \{ \tfrac{1}{2}\} \to S^1 \times \{0,1\}$, right?
You're right, I lied. I was working with $S^1 \times [0, 1]$, not $S^1 \times (0, 1)$. But this is not a big issue, in the sense that there are arbitrarily small tubular neighborhoods so that you can always choose a tubular neighborhood so small that it's closure is topologically $S^1 \times [0, 1]$.
So feel free to replace $S^1 \times (0, 1)$ by $S^1 \times [0, 1]$ throughout.
I agree the question is ill-formatted and the OP is not actively doing anything to make it better: I would have voted to close if I were to vote for anything. But I do not think he said anything offensive: I would have flagged your message too if I had anything to flag.
@BalarkaSen: ok, so you first show that there is a def. ret. from $S^1 \times [0,1]$ minus center circle to $S^1 \times \{0,1\} \subset S^1 \times [0,1]$ and then apply that to get an analogous def. ret. in a surface
@Huy Careful now. We showed that the complement of the chain deformation retracts to the complement of the tubular neighborhood of the chain, note. Now complement of the tubular neighborhood of the chain is connected because it's one of the two connected components of the complement of boundary of the chain.
That implies complement of the chain is connected, is what I am trying to say.
We're not looking at both the components of the complement of the boundary circle. Just one component: the exterior one - complement of the other component (aka the tubular nbhd of the chain)
(and even then, we're pretending we proved this for a chain: we just proved this for a chain of length 1 aka an scc)
@BalarkaSen: the retraction $F(x,1): S^1 \times [0,1] - S^1 \times \{1/2\} \to S^1 \times \{0,1\}$ "pushes" points towards the boundary of $S^1 \times [0,1]$. wouldn't the other direction be the desirable one, here? (because we're moving the boundary of the tub nbhd towards the circle?)
what do you mean by "cutting along tubular neighborhood"? complement of tubular neighborhood of an even length chain is connected because it's a component of the complement of the boundary circle of the tubular neighborhood (along with the other component, namely the tubular neighborhood).
Obviously not, but I'd be a bit hard pushed to write down a proof. I mean, the boundary of the tubular nbhd obviously bounds the tubular nbhd itself :)
So you're essentially asking if the tubular neighborhood is diffeomorphic to all of the complement of the chain. That can hardly ever happen.
I don't see what's particular about "even length" here though.
@Huy you won't, not really. you need to do that, along with a lot of bump function trickeries, to make your deformation smooth, but of course that's not required if you want to prove that # of connected components are the same
just a topological deformation retraction is enough
I guess that brings me back to my original question, being what would the most effective process be for determining the value of x in this instance where a and b values are known. I'm not looking for a solution, more verifying if the processes that I've been using are correct.
(Message 1/3) All users, I add here these calculations, if some user want to do this exercise. I don't know if it is useful this recipe. I start with a trigonometric identity, type for example in Google Books, the words: Murty, Problems in Analytic Number Theory (Springer). Then you can see in page 41, the Exercise 3.2.10.
(Message 2/3) Do the specialization with $\theta=\pi s$ and combine it with the functional equation for the Riemann zeta function. Take the absolute value of previous, and combine it with Proposition 1 of Nazardonyavi and Yakubovich On an inequality for the Riemann zeta-function in the critical strip, this article from arXiV is arxiv.org/pdf/1206.1801.pdf .
