Articles of fundamental groups

Fundamental group obtained by attaching a n-cell with n ≥ 2

I am having trouble with Hatcher’s Algebraic Topology P39, Problem 18: Show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^n$ with $n ≥ 2$, then the inclusion $A \rightarrow X$ induces a surjection on $π_1$. The given hint is to follow the proof of Proposition 1.14, […]

Fundamental Group on Quotient of Unit Disk

I am fairly new to algebraic topology so please bare with me if this seems simple I am trying to find the fundamental group of the unit disk with the identification on the boundary z = (cos(θ), sin(θ)) being mapped to (cos(θ+2π/n), sin(θ+2π/n)). For n=1 it is just the disc so the fundamental group is […]

Does trivial fundamental group imply contractible?

Let $X$ be a path-connected topological space with a trivial fundamental group: $$\pi_1(X,x_0)=\{e\}.$$ Does $X$ have to be homotopic to a point? I know that the converse is true: a contractible space has trivial fundamental group. But what about the converse? Does the fundamental group tells us enough of the space to fix its homotopy […]

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 […]

The Abelianization of $\langle x, a \mid a^2x=xa\rangle$

I wish to verify the following statement (which comes from Fox, “A Quick Trip Through Knot Theory”, although that is probably not important). “$\Gamma=\pi_1 (M)=\langle x, a \mid a^2x=xa\rangle$ so the homology of $M$ is infinite cyclic.” So, I need to find the Abelianization of the fundamental group. Using the relations I get $$y_1:=[x,a]=x^{-1}ax,\qquad y_2:=[a,x]=x^{-1}a^{-1}x$$ […]

How to determine space with a given fundamental group.

I would like to give examples of topological spaces such that their fundamental groups are, respectively, $\mathbb{Z} \oplus \mathbb{Z}_{n}$ and $\mathbb{Z} \ast \mathbb{Z}_{n}$. For the latter, I thought about this: let $X_{n}$ be the space obtained from $S^{1}$ by attaching a 2-cell with characteristic map $\chi (z) = z^{n}$. This map identifies all $n$-th roots […]

Help finding the fundamental group of $S^2 \cup \{xyz=0\}$

let $X=S^2 \cup \{xyz=0\}\subset\mathbb{R}^3$ be the union of the unit sphere with the 3 coordinate planes. I’d like to find the fundamental group of $X$. These are my ideas: I think the first thing I should do is to retract all the points outside the sphere to the sphere (is that possible? how?) then using […]

Fundamental group of quotient of $S^1 \times $

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$. I’ve been trying to visualize $X$ as a cylinder of height 1 with the two ends identified `with a […]

Finding a space with given homology groups and fundamental group

I started thinking a couple days ago about the example below, and it led me to ask the following question: How (or when?) can we build a space (let’s say a CW complex) with given homology groups and fundamental group? I know we can do either of these separately, but I couldn’t do both simultaneously […]

Why is the fundamental group a sheaf in the etale topology?

In this paper by Minhyong Kim on p5, there is a variety $X$ defined over $\mathbb{Q}$, $G = \pi_1(X(\mathbb{C}),b)$ the topological fundamental group of the associated complex algebraic variety, and $G$^ the profinite completion of $G$. Kim states that $G$^ is a sheaf of groups for the etale topology on Spec($\mathbb{Q}$). Why is this? A […]