# Tagged Questions

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

102 views

397 views

### Surface has Euler characteristic 2 iff equal to sphere

Let $\Sigma$ be a connected (not necessarily compact) surface with or without boundary. Is it true that $\Sigma$ is homeomorphic to the sphere if it has euler characteristic $\chi(\Sigma)\geq 2$? I ...
115 views

### Definiton of Submanifold of Topological Manifold

Analogy to smooth manifold, I want to define the submanifold of topological manifold. There are two ways. Let $M$ be a topological manifold, and $N\subset M$. If $N$ is a topological manifold, then ...
51 views

### Relation between Map and Dimension

I am curious about two questions below Let $M$, $N$ be two topological manifold. If $\dim M>\dim N$, is there exist an injective continuous map $f: M\rightarrow N$? If $\dim M<\dim N$, is ...
27 views

159 views

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group $... 2answers 50 views ### Charts for level set manifolds & Multiplication map$F(A,B)=AB$from$O(n)\times O(n)\to O(n)$is smooth This is homework so no answers please We have Multiplication map$F:O(n)\times O(n)\to O(n)$defined as$F(A,B)=ABF:O(n)\times O(n)\to O(n)$, where$O(n)=\{A\in M(n\times n):AA^{t}=id\}$. The ... 1answer 126 views ### Atlases on the topological manifold$\mathbb R$I have been trying to produce an example of two incompatible atlases on$\mathbb R$. But no success. Could someone help me please? All my example seem compatible. For example,$A_1 = \{((-\infty,1), \...
I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaystyle\bar f>> E'\\ ...