Difference between revisions of "Topologie - 2019"

Revision as of 13:30, 13 August 2019

Andrea, 12.08.2019 10:50-11:10

Topics:

• What is the Fundamental group of the Möbius strip?

When two spaces have the same Fundamental Group? Def of Homotopy equivalent spaces and find the homotopy equivalence between the cirle and Mobius strip.

• exercise: you have (xn) sequence with limit x. Prove that the set containing only the sequence and the limit is compact.

I took a long time for the first part and unfortunately after there was no time left.

Sisto is calm and helps you if you make confusion.

Nico, 13.08.2019, 11:30-11:50

Questions:

• Let \(X\) be a compact metric space. Suppose that \( f: X \rightarrow Y \) is continuous and \( f(x) \neq x \) for all \( x \in X \). Prove that there exists \( \epsilon > 0 \) so that \( d(x,f(x)) >\epsilon \) for all \( x \in X \).
• Let \( f , g : X \rightarrow Y \) be homotopic maps, \( X , Y \) topological spaces. Show how the induced map of \( f \) and \( g \) are related.

I needed more than half of the time for the first questions where I couldn't conclude at the end. He tried to help me and sometimes he let me think for a couple of seconds without saying something. We run out of time before I finished everything of the second question.

In front of his office is a couch with some chocolate, one could eat. That is really nice.

Viera, 13.08.2019, 13:40-14:00

Questions:

• Let \(A \subseteq X \) be a discrete subset of a compact space. Is A finite? If not, which additional condition makes it true?
• Give two examples of covering of the circle.

After the 2nd question, he wanted another example of a covering. He moved on to ask me, what happens on the level of fundamental groups. He wanted to know, whether there is a covering, where the fundamental group is mapped to the trivial group (this would be \(\mathbb{R}\)), and then asked if there is also a compact space with this property. However, Luca stopped him and told him, this would be a too hard question, so we stopped at this point.

Tobias, 13.08.2019, 14:20-14:40

Sisto says hi, you give Luca your Legi and you start (really no smalltalk at all). You have a stack of paper and pens ready.

• Let \(X\) compact, \(A \subseteq X\) a discrete subspace. Is \(A \) finite? Which conditions are additionally needed?

I was quite nervous at the beginning and had some problems despite knowing the anserwer that \(A\) should be closed. After some time I came up with the solution.

• Do you remember roughly how we proved that \(X\) compact, \(A \subseteq X\) closed implies \(A\) compact?

I explained the idea of the proof in two sentences without writing something down.

• Let's do something from the second part of the lecture. Sisto lets Luca choose the fundamental group of the Klein Bottle.

I started explaining van Kampen, chose sets A and B as in the lecture but was most of the time stuck trying to show that the fundamental group of the outer set was \( R*R \)

• What is the structure where we faced free groups?

I drew the two rings which intersect at one point.

• Are these homeomorphic?

Yes. Then I though again about the question and noticed that they only have the same Fundamental group but are probably not homeomorphic.

• How do you show they have the same fundamental group?

I came with the idea of homotopy equivalence but didn't really know where to start. He gave me some hints how to choose f and g. The time was already over, I explained with one sentence how I get the condition \( abab^{-1} = 1 \).

Sisto waits quite long with giving hints. So take your time, don't get nervous.