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Mar
27
comment A Proof of Bruns' Theorem
Sorry, @Shahab, I don't remember why I asked this question or in what context I found the theorem.
Mar
18
comment Why are two permutations conjugate iff they have the same cycle structure?
I don't remember, but you can look through my questions, @Liebe.
Mar
16
accepted Lie derivative of a two-form
Mar
16
comment Lie derivative of a two-form
Take Cartan's formula if you will, Mr. @Albanese.
Mar
16
asked Lie derivative of a two-form
Jan
23
accepted The normal curvature is bounded by the principal curvatures.
Jan
18
revised Induced maps of the circle exponents
added 127 characters in body
Jan
18
accepted Induced maps of the circle exponents
Jan
18
comment Induced maps of the circle exponents
I apologize. After going back to the texts I realized that I am describing the induced map on singular homology and this is indeed what I want to compute. I want to compute the winding number of the map that winds around the circle $n$ times. Thank you for your patience.
Jan
16
comment Induced maps of the circle exponents
You claim that $f_n$ is homotopic to the simplical map $g_n$ between $S^1$ and $S^1$ that sends the images of the simplexes in $T$ to these in $S$ just around the circle. Intuitively this must be so, but then again intuitively already knew that the winding number of $f_n$ is $n$. I'm sorry, I cannot prove that $f_n$ and $g_n$ are homotopic.
Jan
16
comment Induced maps of the circle exponents
I'm not claiming that $f_n$ is simplical. I encountered this notion in your answer for the first time. I should assume that by simplical you mean a map sending a simplex to another simplex. Moreover, I explained already how $f_n$ induces a chain map. This map factors down to the level of homology.
Jan
15
comment Induced maps of the circle exponents
I'm sorry, this is not the issue. We can consider a triangulation to be the union of the images of the simplixes. Really, how can we see that $f_n$ is homotopic to your simplical map?
Jan
15
comment Induced maps of the circle exponents
You see now why I don't understand your claim that $f_n$ is homotopic to a simplical map. To me a simplex is a map from the canonical simplex to the topological space in question. A triangulation with $3n$ edges is a collection of $3n$ 1-simplices, i.e. they are $3n$ maps $s:[0,1]\to S^1$.
Jan
15
comment Induced maps of the circle exponents
Any continuous map $f$ induces a homomorphism on homology groups. First, we define the map $f_#$ sending simplexes to simplexes via composition. $f_# (s) = f\circ s$. We extend $f_#$ to sums of simplexes by linearity. Moreover, $f_#$ commutes with the boundary operator, i.e. it defines a chain map. Therefore, $f$ induces a homomorphism of homology groups.
Jan
14
comment Induced maps of the circle exponents
Thank you, sir. What do you mean when you say that $f_n$ is homotopic to a simplical map $T \to S$? Two functions can be homotopic only if they have the same domain and the range of one is a subset of the range of the other, can they not?
Jan
13
asked Induced maps of the circle exponents
Nov
6
awarded  Popular Question
Jul
27
accepted Lifting a principal G-bundle to a principal bundle with structure group a covering of G
Jul
27
comment Lifting a principal G-bundle to a principal bundle with structure group a covering of G
I read that both Stiefel-Whitney classes $w_1$ and $w_2$ need to vanish for a spin structure to exist. Where may I read a proof of this?
Jul
27
asked Lifting a principal G-bundle to a principal bundle with structure group a covering of G