Articles of algebraic topology

Covering spaces of wedge sum of circles

Let’s suppose I’d like to characterise all connected covering spaces of wedge sum of circles.When it comes to a torus, a sphere or $ S^{1} \vee S^{2} $ it is easy to point out exactly how do all covers look like.All I know is how to construct universal covering of $S^{1} \vee S^{1} $ and […]

Hairy Ball Theorem: homotopy from the identity to the antipodal map

I’m having trouble to prove the following statement: If the $n$-sphere $S^n\subset\mathbb{R}^{n+1}$ admits a (continuous) nonvanishing tangent vector field, then $n$ is odd. The idea of the proof is pretty clear on my mind: we use the fact that the antipodal map has degree $(-1)^{n+1}$ and that the Brouwer degree is invariant under homotopies. We […]

Why is homology invariant under deformation retraction/homotopy equivalence?

This is pretty basic, but I can’t find any answers on this site on this already. I want to know why the homology of two spaces $X$ and $Y$ are the same when $X$ deformation retracts/is homotopy equivalent to $Y$. I would appreciate an intuitive explanation and a formal proof. Specifically I’m working through Hatcher’s […]

Query about proof of homotopy lifting property

Let $p : Y \to X $ be a covering projection. Given a path $ u : I \to X $ and a point $ y \in Y $ with $p(y) = u(0) $, there exists a unique path $ \hat{u}:I \to Y $ with $p \hat{u} = u$ and $ \hat{u}(0) = y$. The […]

$H_2$ and $\pi_1$ of open subsets of $\mathbb{R}^2$

If $U$ is an open subset of $\mathbb{R}^2$, is it true that $H_2(U)=0$ and what can we say about $\pi_1(U)$? (For example, can we show that $\pi_1$ isn’t perfect, e.g. $\pi_1(U)\neq 0\Rightarrow H_1(U)\neq 0$?) For $\pi_1$, I’m aware that there is an answer here: fundamental groups of open subsets of the plane, but I’d rather […]

What is the definition of “sheet”?

Here is a definition for evenly covered set. Definition Let $C,X$ be topological spaces and $p:C\rightarrow X$ is a continuous function. Let $U$ be an open subset of $X$. Then $U$ is said to be evenly covered by $p$ if there exists a family $\{V_i\}_{i\in I}$ of mutually disjoint open sets in $C$ such that […]

How to show the covering space of an orientable manifold is orientable

I’m trying to prove this using purely topological arguments, no differential geometry as I haven’t been exposed to it. I’ve been playing around with definitions a bit and here’s what I have so far. Let $M$ be an orientable manifold. Let $N$ be its covering space. Then we have an orientation function $\mu : M […]

Basic constructions for graded algebras.

I’m reading about the Weil algebra of a Lie group and it involves some constructions I’m not very familiar with, for instance the “free graded-commutative graded algebra on $a_1…a_n$ with degrees $deg(a_i)$.” Does anyone know a good source for basic graded-algebra constructions like this? Anything i find is either too basic or too advanced to […]

How does a surjection from $\pi_1(\Sigma_g)$ to a group $Q$ determine an element of $H_2(Q,\mathbb{Z})$?

Let $\Sigma_g$ be a closed orientable surface of genus $g$ on page 36 of this paper. It is asserted that for a finite group $Q$, a homomorphism $\pi_1(\Sigma_g)\rightarrow Q$ determines an element in $H_2(Q,\mathbb{Z})$. How does this work? This is especially confusing to me, since they don’t even specify the action of $Q$ on $\mathbb{Z}$, […]

Does there exist a $C^1$-path path-homotopic to a rectifiable curve?

Related: Let $G$ be an open connected subset of $\mathbb{C}$. Let $\gamma:[0,1]\rightarrow G$ be a rectifiable curve. Then, does there exist a $C^1$-curve $\Gamma:[0,1]\rightarrow G$ such that $\gamma$ and $\Gamma$ are homotopic relative to $\{0,1\}$ in $G$?