Recent content by PsychonautQQ

I want to understand all possible homomorphisms ##\alpha: Z^a -> Z^b## as well as understand what a matrix representation for an arbitrary one of these homomorphisms would look like. Furthermore, under what conditions does a homomorphism have a matrix representation? To begin, let...

Hey PF! This isn't for homework, just me messing around with some thoughts in caluclating various homology groups. So suppose we have ##p \in S^n## and suppose that ##X## is a Polyhedra. I want to show that ##H_q(X \times S^n, X \times p) \cong H_{q-n}(X)## I was given the hint to start out...

Thank you. Even with your brilliant insight's, I have failed to proceed with the computations :-(.

Technically speaking, the problem that I'm working on involves taking the complement of a tube around a knot (an embedding of a circle into ##R^3##) and calculating the homology group of this space. The approach that I'm using is to use a mayer-vietrois sequence. So let ##K## be a knot and let...

This isn't homework, it's a proof left to the reader as I self study Munkre's 'Elements of Algebraic Topology' Prove that if the sequence ##A_1 --> A_2 --> A_3 --> A_4 --> A_5## is exact Then so is the induced sequence: ##0 --> cok(a_1) --> A_3 --> ker(a_4) --> 0## where ##a_1## and ##a_4##...

Can someone help me to understand what the boundary operator on a p-chain is doing exactly? Or boundary operators in general? I really need to develop a better intuition on the matter.

Suppose ##q: E-->X## is a covering map (not necessarily normal). Let ##E' = E/ Aut_{q}(E)## be the orbit space, and let ##\pi: E-->E'## be the quotient map. Then there is a covering map ##q': E' --->X## such that ##q' * \pi = q## where ##*## is composition of functions. I am confused why ##E'##...

Can you explain why all points who's fibers have cardinality ##abs(/lambda)## is open?

I understand. Is it possible to 'cut' the torus in the way depicted in the diagram and wind up with a topologically equivalent space that would then be two Klein bottles?

10/10 for deciphering my garbage and giving a great response anyway

Why are the two pieces not Klein bottles? It looks like the orientations of the edges are set up correctly to me.

I'm having trouble following one part of a proof. Proposition: For any covering map ##p: X-->Y##, the cardinality of the fibers ##p^{-1}(q)## is the same for all fibers Proof: If U is any evenly coverd open set in ##X##, each component of ##p^{-1}(q)## contains exactly one point of each fiber...

Wow, thanks for the great response! Can you explain how the observation of seeing two klein bottles in this diagram leads to the observation to there is an n-sheeted covering map?

So I'm trying to understand how the Torus is a 2-sheet covering of the Klein bottle. I found this on math exchange: https://math.stackexchange.com/questions/1073425/two-sheeted-covering-of-the-klein-bottle-by-the-torus. The top response add's rigor to of the OP's observation that the double...

N union an open set contained within the boundary of the disk we remove from M when forming the connected sum?