# Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

11 views

### Isomorphic homotopy groups of universal cover?

In Ralph Cohen's notes on the topology of fiber bundle he says (1) on pp.167, $BSO(n) \to BO(n)$ is a universal cover thus $\pi_i(BSO(n)) \to \pi_i(BO(n))$ is an isomorphism for $i \geq 2$ (2) on pp....
17 views

### Why is topological K-theory equivalent to nonabelian cohomology with respect to the stable unitary group?

I was reading on the $n$Lab page for topological K-theory that taking cohomology of a smooth space with respect to the smooth $\infty$-stack $\mathbf{Vect}$ is equivalent to taking its cohomology with ...
27 views

### category-theory, right group action

Let $G$ be a group. We observe the category $(Set)_G$ of right group actions. a) Let $X\times G\to X$ and $Y\times G\to Y$ be two transitive right group actions with $x\in X$ and $y\in Y$. Find a ...
393 views

35 views

### How to construct a homotopy equivalence between a mobius band and a circle?

A mobius band is homotopic equivalent to a circle because the mobius band can deformation retract onto a circle. I am wondering how could we understand this fact from the definition of being ...
23 views

### Proof of Brouwer fixed point theorem using change of variable formula. [on hold]

Is there a proof of Brouwer fixed point theorem using change of variable formula for integration?
38 views

79 views

42 views

### $H_{k+1}(X \cup_f D^{k+1},X) = ?$

I am stuck with the calculation of the following homology group: $H_{k+1}(X \cup_f D^{k+1},X) = ?$ where $X$ is a simply-connected CW complex and $f: S^k \to X$ is a continuous map (attaching map of ...
49 views

### Which group homomorphisms induce the action of the fundamental group on the fiber?

Given topological spaces $X$ and $Y$ and a covering map $p: X \rightarrow Y$, we know that the group $\pi_1(Y,y_0)$, where $y_0\in Y$, acts on the fiber $F=p^{-1}(y_0)$. Also, we know that the set ...
16 views

### Seifert matrix, linking numbers, generators

I have been asked to compute the seifert form of a knot, the twist knot. I know how to compute the seifert surface, and then the seifert matrix seems to be defined accordingly (according to all the ...
40 views

### Homology group of an open set on $S^1$

Let $U$ be an open set which is constructed as intersection of $S^1$ and open ball in $\mathbb{R}^2$. And $x$ is just a point contained in $U$. My opinion: By long exact sequence, $H_n(U, U-x)$ is ...
32 views

37 views

### The Incluson Map $S_1\to S_1\times S_1$ Induces an Injection in First Homology.

Let $T=S^1\times S^1$ be the torus and $A=S^1\times \{x_0\}$ be the "vertical" circle in the usual depiction of the torus as a tyre tube sitting "horizontally". Let $i:A\to T$ be the inclusion map....
This is a modification of the question I previously asked here. Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting ...