Mathematics

Homology sphere A homology sphere is a (closed connected oriented) $n$-mainfolds with the same homology as the $n$-sphere. Poincaré first conjectured that any $n$-manifolds homologous (i.e. having the same homology) to the $n$-sphere should be homeomorphic to the $n$-sphere, then later he found a counterexample, which led him to a modified conjecture (which is the Poincaré’s conjecture). The counterexample he found is very interesting, one of the way to construct it is the folowing:...

Using fields to detect integral homology? So in the universal coefficient theorem, by taking $R = \mathbb{Z}$, we know that there is a short exact sequence (which splits): $0 \rightarrow H_n(C_{*}) \otimes_R M \rightarrow H_n(C_{*} \otimes_R M) \rightarrow Tor^R_1(H_{n-1}(C_*), M) \rightarrow 0$ Some interesting cases are when $M = \mathbb{Z}_{p} \text{ or } \mathbb{Q}$ , do the homology over $\mathbb{Q}$ and $\mathbb{Z}_p$ tells us something about the integral homology? In particular, if $\tilde{H}(X; \mathbb{Q}) = 0$ and $\tilde{H}(X; \mathbb{Z}_p) = 0$ for all $p$, does it imply that $\tilde{H}(X) = 0$ ?...

Stone–Čech compactification One of the proof of the Stone–Čech compactification is to consider the continuous functions from a given topological space $X$ to $[0, 1]$. Crux 1: constructing $[0, 1]^{C}$ Let $C$ be the space of all continuous functions from $X$ to $[0, 1]$, consider $[0, 1]^{C}$, there is a natural map from $X$ to $[0, 1]^{C}$: for each $x \in X$, define $\phi(x) = f \mapsto f(x)$. With product topology $\phi$ is continuous....

I was going through a problem in Lee’s Topological Manifolds book1 (problem 7-12). It states that the infinite broom has a strong deformation retract to $(0, 0)$ but not for $(1, 0)$ (only a deformation retract). Below is some of my playful observations inspired by this problem. (Not restricted to the infinite broom space.) Inifinite Broom Note that the deformation retract to $(0, 0)$ itselfs imply the deformation retract to $(1, 0)$, since one can define the deformation retract to $(1, 0)$ by “first retract to $(0, 0)$, then push it to the point $(1, 0)$ along the line joining these two points”....

I am reviewing some fundamental algebra, and I just learnt something beautifully suggesting the connection of ring theory and geometry, which made me eager to learn some algebraic geometry (which I was not very interested when I was in undergrad). It is an exercise in Aluffi’s book1. Lets’ jump into it. The problem Let $K$ be a compact topological space, and let $R$ be the ring of continuous real-valued functions on $K$, with addition and multiplication defined pointwise....