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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