Articles of algebraic topology

How to show that a diagram is a pushout in the category $\text{TOP}$?

Show that the diagram: $$\require{AMScd} \begin{CD} A @>f>> Y \\ @ViVV @VVV \\ X @>>> X \amalg_f Y \end{CD}$$ where $i : A \hookrightarrow X$ is an inclusion is a pushout in $\mathbf{Top}$. I can prove that $g_1 : Y \rightarrow X \coprod_fY$, and $g_2 : X \rightarrow X\coprod_f Y$ is a solution of the […]

Geometrical interpretation of simplices

I’ve just started doing simplices, where the $n$-simplex has been defined to be $$\Delta^n = \{x \in \mathbb{R}^{n+1}\mid x_i \geq 0, \sum x_i=1\}.$$ It’s easy to see that the $0$-simplex is the point $1$ in $\mathbb{R}^1$, the $1$-simplex is the line from $(1,0)$ to $(0,1)$ in $\mathbb{R}^2$, and the $2$-simplex is the triangle, including the […]

Is a simple curve which is nulhomotopic the boundary of a surface?

Let $C$ be a simple curve in an open subset $U$ of $\mathbb R^3$. Suppose that $C$ is nulhomotopic in $U$. Must there exist a homeomorphism $f$ from the closed unit disk $D$ in $\mathbb R^2$ to $U$ such that $f(\partial D) = C$? This seems intuitively like it should be true, and I believe […]

Practicing Seifert van Kampen

I am just practicing how to use Seifert-van Kampen. The following excercise is from Hatcher, p.53. Let $X$ be the quotient space of $S^2$ obtained by identifying the north and south poles to a single point. Put a cell complex structure on $X$ and use this to compute $\pi_1(X)$. I found a cell complex structure […]

Are all paths with the same endpoints homotopic in a simply connected region?

It is clear to me that if all paths (with the same endpoints) in a region are homotopic then that region is simply connected, however I am having difficultly proving the converse, that is, all paths with the same endpoints are homotopic in a simply connected region. Here is what I have so far Given […]

Is the product of covering maps a covering map?

I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times X_2 \rightarrow Y_1 \times Y_2$ a covering map?

$H$ a discrete subgroup of topological group $G$ $\implies$ there exists an open $U\supseteq\{1\}$ s.t. the $hU$ are pairwise disjoint

Problem: Let $G$ be a topological group (i.e., $G$ is a topological space and $G \times G\rightarrow G$ is continuous), and let $H$ be a discrete subgroup of $G$. Prove that there is a neighborhood, $U$, of the identity, $1$, such that the sets $h \cdot U$, $h \in H$ are pairwise disjoint. Hint: First […]

Existence of a universal cover of a manifold.

Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don’t come along with this, like a global Hausdorff condition mentioned here, second countability or paracompactness etc. Mainly it would seem to rule out certain pathological […]

On the definition of projective vector bundle.

Definition 1 : Let $B$ be a topological space. A complex vector bundle of rank $n$ over $B$ consists of the data $(E,B,\pi,\{U_i\}_i,\{\phi_i\}_i)$ where $E$ is a topological space, $\pi:E\to B$ is a continuous map called the projection map, $\{U_i\}_i$ is an open cover of $B$ and $\phi_i:\pi^{-1}(U_i)\to U_i\times \mathbb C^n$ is a homeomorphism such […]

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted covering by an orientable one of genus n-1. I tried to use the polygonal representation of these surfaces and try to get one […]