I asked Deepseek to draw the Schlegel diagram of a hexahedron — should be a very simple task right? Just 2 squares and link up their vertices. And yet I noticed something interesting: it tends to overthink and end up outputting nonsense answers when DeepThink is on (which is supposed to be more powerful and “clever”), but getting the task done correctly when DeepThink is off. Huh. So this is the prompt I gave Deepseek:...
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 $\tilde{H}(X) = 0$ ?...
That’s why I love Arnold’s books.
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....