# Tag: algebraic

## algebraic topology – Homotopy between inverse path

I’m really struggling with some exercise my professor left me about fundamental group so I think I need some clarification.

During one of them I found that any loop $omega$, where $omega$ belongs to the fundamental group of a connected space, it is homotopic to its inverse $omega^{-1}$, but what does that means?

In my opinion the only way that could happen is that they are both contractible. Am I missing any other possibility?

## Solve gets stuck with rater simple algebraic equation

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## ag.algebraic geometry – Degree of polynomials describing projection of algebraic set

Consider an algebraic subset $Vsubseteq mathbb{R}^{n+1}$ defined as the zero set of polynomials ${f_i}$ and the projection map $pi: mathbb{R}^{n+1}to mathbb{R}^n$ deleting the last entry.

By the Tarski-Seidenberg theorem, the image $pi(V)$ is a semi-algebraic set. Is there a bound on the degree of the polynomials characterizing $pi(V)$ in terms of the degrees of the polynomials $f_i$?

## Tips needed for speeding up solving system of nonlinear algebraic equations

`**

**`I am Vedat.I have trouble with the attached code.

In this code I solve system of large number of nonlinear algebraic equations by using FindRoot command.

The only thing I need is a fast running program since

my code runs very slowly for the small values of h,for example h=0.005 or 0.001.

Your comments are highly welcome. Thank you.

## algebraic topology – Methods for determining the degree of a map

What methods can I use to solve the following question?

Compute the degree of the map $f : S^1 rightarrow mathbb R$, $f(theta) = sin(2theta)$?

I know that $H_1(S^1) cong mathbb R$, so it feels like the answer should be 2, because of the doubling of the angle, but I don’t know how to compute it. I’ve heard that you can use the Künneth formula, but I haven’t seen any concrete examples of applying it to solve such a question.

## algebraic topology – Homology group $H_*(Xtimes S^n)$

Given any topological space $X$, I want to compute the homology group of $H_*(Xtimes S^n)$ by using Mayer-Vietoris sequence (**NOT Knnuth formula**). Here is what I have so far:

When $n=0$, it’s clear the result is simply $H_*(X)$.

For $ngeq1$, let $A, B$ be the upper half and lower half of $S^n$, then we have $S^n=Acup B$ with $Asimeq D^n$, $Bsimeq D^n$ and $Acap B simeq S^{n-1}$. Using this open cover of $S^n$, we get an open cover of $Xtimes S^n$ by $Xtimes S^n=(Xtimes A) cup (Xtimes B)$ with $Xtimes A simeq Xtimes D^n simeq X$, $Xtimes Bsimeq Xtimes D^nsimeq X$ and $(Xtimes A) cap (Xtimes B)=Xtimes (Acap B) simeq Xtimes S^{n-1}$. Apply Mayer-Vietoris theorem, and notice that $X, Xtimes S^{n-1}, Xtimes S^n$ are all connected, we then get an exact sequence:

$$0 xrightarrow{partial_*} H_m(Xtimes S^{n-1}) xrightarrow{i_*oplus j_*} H_m(X)oplus H_m(X) xrightarrow{k_*-l_*} H_m(Xtimes S^n) xrightarrow{partial_*}$$

$$ H_{m-1}(Xtimes S^{n-1}) xrightarrow{i_*oplus j_*} H_{m-1}(X)oplus H_{m-1}(X) xrightarrow{k_*-l_*} H_{m-1}(Xtimes S^n) xrightarrow{partial_*} …$$

$$… xrightarrow{partial_*} H_1(Xtimes S^{n-1}) xrightarrow{i_*oplus j_*} H_1(X)oplus H_1(X) xrightarrow{k_*-l_*} H_1(Xtimes S^n) xrightarrow{partial_*}$$

$$mathbb Z rightarrow mathbb Zoplus mathbb Zrightarrow mathbb Z rightarrow 0$$

(The notion of $i,j,k,l$ are standard, just the notation used in Wikipedia)

This is the first time I meet a problem that can not be dealt only with homological structure, that means it should involves the detail analysis of those homomorphisms induced by inclusion maps. But I’m totally missing in their analysis, for example, in my view, $k_*$ should be equal to $l_*$ since we are essentially dealing with same topological objects here, but this seems not to be true… Can anyone give a description of those homomorphisms involved in the sequence? This will be very instructive for my study, I would really appreciate!

## algebraic topology – Why the degree of a proper map is well defined?

Consider a proper map $f:M=mathbb{R}^nto N=mathbb{R}^n$, we define the degree of it as followes.

Choose a generator $omegain H_c^n(N)$, then $int_Nomega=1$, we define $deg f=int_Mf^*omega$ where $f^*$ means pullback.

I don’t know why this is well defined. Moreover, if we consider

Then this seems right. But if $f$ is not surjective, then we let $V=N-mathrm{Im}f$. If we have two forms $mathrm{supp}alphasubset V,mathbb{supp}betasubsetmathrm{Im}f$ with $intalpha=intbeta=1$, then

$$int f^*(alpha-beta)=int f^*alpha-int f^*beta=-int f^*betaneq0,$$

but $int(alpha-beta)=0$, so we should have

$$int f^*(alpha-beta)=deg fint(alpha-beta)=0,$$

which is impossible.

## algebraic geometry – Why do tensors have a generic rank?

There is an open dense set of constant rank tensors in $mathbb{C}^{n_1} otimes mathbb{C}^{n_2} otimes ldots otimes mathbb{C}^{n_d}$. Intuitively, this means that if the entries of a hypermatrix are chosen by iid uniform random variables, with probability 1 the hypermatrix will all have rank $r$ for some constant $r$. It seems intuitive for there to be a generic multilinear rank, since choosing the entries of a matrix randomly and independently should almost never result in a relation between the rows or columns of a matrix. However, the existence of a generic rank seems remarkable and even counterintuitive. Is there any way to look at tensor rank so that generic rank seems intutive?

Also note that there is no generic rank in general for tensors over $mathbb{R}$, and the generic rank over $mathbb{C}$ is in general not the maximum possible rank, which is precisely why there are tensors with no best low rank approximations.

## Using algebraic geometry to understand class field theory

In *Algebraic Number Theory*, S. Lang says “(a geometrical approach) allows one to have a much clearer insight into the whole class field theory, since the existence theorem and

the reciprocity law become obvious once the machinery of algebraic

geometry is available.”

Inspired by this, I wonder if there is some (preferably modern) reference for class field theory using algebraic geometry.