Mathematics

12:44 AM

@shintuku I know how to relate arithmetics modulo $n$ and modulo $m \gt 2(n-1)$. With a polynomial!

What you get is a polynomial $f \in \Bbb{Z}[X]$ such that $f(x + y) = f(x) +_n f(y) \pmod m$ for all $x, y \in \{0, \dots, n-1\}$.

So it's as if the structure $(\Bbb{Z}/n ,+)$ sits as a polynomial image inside of the abelian group $(\Bbb{Z}/m, +)$. For divisorially unrelated $m,n$ that is usually unheard of to relate them!

0

Q: Trick to have a polynomial relationship of $(\Bbb{Z}/n, +)$ with $(\Bbb{Z}/m, +)$ for $2 \leq n$ and $m \gt 2(n-1)$?

Define: $$ g(z) = (z(z - 1)(z-2) \cdots (z-n + 1))^{\varphi(m)-1} \equiv \begin{cases} 0 \text{ if } 0\leq z \leq n \\ 1 \text{ otherwise}\end{cases} \pmod m $$ We can use $g(z)$ to define polynomially instead of piece-wise the following map: $$ f(z) := \begin{cases} z \text{ if } g(z) = 0 \\ z ...

one potato two potato

1:19 AM

Let $G$ be a group and $X$ be a $G$-set. If $F$ is a forgetful functor from the
category of $G$-set to the category of set, then we can define a bijection $(\eta^g)_X = \varphi_X(g):F(X)\to F(X)$ where $\varphi_X:G\to S_X$ is a permutation representation of $X$. If $\alpha:F\to F$ is a natural isomorphism, then I'm trying to show $\alpha = \eta^g$ for some $g$.

So I need to find some suitable $g$. If $X = G$, considering $G$ as a $G$-set with left-multiplication, then $\alpha_G:F(G)\to F(G)$ so $\alpha_G(1) = g$. Now I'm hoping $\eta^g = \alpha$.

Is this correct? I can't proceed from here. Naturality of $\alpha$ can be used to show that $\alpha_X:F(X)\to F(X)$ is a $G$-equivariant map I think but don't know what to do next.

geocalc33

@DLeftAdjointtoU what's up

Koro

2:25 AM

4

Q: $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$

I've been doing some more exercises in Hatcher, in particular the following: Show that $H_0(X,A) = 0$ iff $A$ meets each path-component of $X$. "$\Leftarrow$": Let $x_i \in A \cap X_i \neq \emptyset \forall $ path-components $X_i$. Then for any $x \in X_i$ there is a path $\gamma$ from $x_i$ to...

algebraic-topology

How to do this? Any ideas?

I don't understand the answer there at all.

D Left Adjoint to U

2:42 AM

@geocalc33

hey

Koro

0

Q: $H_0(X,A)=0$ iff $A$ intersects every path component of $X$.

We have a long exact sequence $...\to H_0(A)\to H_0(X)\to H_0(X,A)\to 0$ If $H_0(X,A)=0$, then by exactness at $H_0(X)$: Image $(H_0(A)\to H_0(X))=H_0(X)=$ ker $(H_0(X)\to H_0(X,A))$ so the map $i_*:H_0(A)\to H_0(X)$ is surjective. Let $X_a$ be a path component of $X$. Suppose on the contrary tha...

algebraic-topology homology-cohomology exact-sequence

leslie townes

3:06 AM

koro, given the relation between your question and the earlier one, it might make sense to link to the previous question in your post (maybe with commentary about what you don't understand about the earlier answer, or how your question is different). if only to avoid having someone respond by referring you there or marking it as a duplicate.

one potato two potato

I remember it's one of the exercise problem in hatcher

Koro

3:21 AM

I don't understand how surjectivity of the map is related to path connectedness.

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$Mathematics$ Mathematics