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Questions tagged [algebraic-topology]

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

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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Basic Algebraic Topology puzzler

I've been watching Norman Wildberger's lectures on Algebraic Topology and one of his problems really got me stuck. The question is to show how a double-holed torus with a line of infinite length passing through one of the holes is homeomorphic to a…
Connor James
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How to prove every non-compact, connected 2 dimensional surface is homotopical to a bouquet of flowers?

This is one of my old unsolved questions when I reading Novikov's book on homology theory. I do not know how to prove it because standard triangulation, fundamental diagram, etc does not help and it should be easy to prove.
Bombyx mori
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Realization of graded algebras with Poincaré duality

Question: Given a finite dimensional positively graded algebra $A$ over some ring $R$ that satisfies Poincaré duality in some dimension $n$, is there necessarily a topological space $X$ such that $H^*(X;R) \cong A$? I recognise this is some…
AnonymousCoward
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Compute the homology groups using Mayer-Vietoris sequence

I need to compute the homology groups under group of integers $ H_k(D; \mathbb Z) $, of the simplicial complex being a triangulation of the following figure: I divided it two parts $D = L_1 \cup L_2$ like this: Then both parts $L_i$ are…
Stanislav Morozov
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triangulating torus using simplices

i am currently pursuing a course in basic homology theory.i am really stuck in how to triangulate a torus by using simplicial complexes . in every book, a diagram is given but it does not define that why are we subdividing the sheet into a no of…
Abhishek Shrivastava
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Is this kind of simplicial complex necessarily homotopy equivalent to a wedge of spheres?

Suppose $d \ge 2$ and $S$ is a finite simplicial complex of dimension $2d$, such that (1) $S$ is simply connected, i.e. $\pi_1 ( S) = 0$, and (2) all the homology of $S$ is in middle degree, i.e. $\widetilde{H}_i ( S, \mathbb{Z}) = 0,$ unless $i =…
Matthew Kahle
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Classify Open Sets in $\mathbb R^2$

In $\mathbb R$, we know that connected open set is $(0,1)$ under homeomorphism. I am wondering what is the situation in $\mathbb R^2$. From $\mathbb R^2-\text{pt}\simeq S^1$, we will have two open sets $\mathbb R^2$ and $\mathbb R^2-\text{pt}$.…
gaoxinge
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Direct sum of cohomology rings. How to interpret?

Consider the space $Y:= M(\mathbb{Z}_p, 2) \vee S^4)$ where $M(\mathbb{Z}_p, 2)$ is a Moore space, i.e having trivial homology groups for all $i \neq 0,2$ where $H_0(M) \cong \mathbb{Z}$ and $H_2(M) \cong \mathbb{Z}_p$. Hatcher gives me the…
user7090
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CW structure of lens space

I am reading page 144 from hatcher's book about lens space, but I am really confused about the CW-structure described there. He first takes one unit circle from one factor of $\mathbb C^n$, then divide it into m parts. Then he join jth vertex to…
user198206
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If $p:(E,e_0) \to (X, x_0)$ is a simply connected covering space with group of covering transformations $G$ then $G \cong\pi_1(X,x_0)$.

In this theorem, to prove that $\chi$ is onto they construct the function $\phi$ such that $\chi(\phi)=[\sigma]$. They have to prove $\phi \in G$, so they prove $\phi$ is continuous. But in order to have $\phi \in G$ we also need that it's an…
user29232
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Fundamental group of $X = \{(p, q)|p \neq −q\}\subset S^n \times S^n$

This problems appears in Chapter 2, exercise 3 from "A Concise Course in Algebraic Topology J. P. May" book. Let $X = \{(p, q)|p \neq −q\}\subset S^n \times S^n$. Define a map $f:S^n\to X$ by $f(p) = (p, p)$. Prove that $f$ is a homotopy…
Gaston Burrull
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$\pi_7(S^4)$ contains an element of infinite order.

Show that $\pi_7(S^4)$ contains an element of infinite order. Now, I know that I should probably use the Hopf bundle here somewhere. I also know that $\pi_3(S^7) = 0$. But I am stuck. Can anyone help me?
Math
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Integral homology of $S^{n-1}/\pi$, $H_*(S^{n-1}/\pi; \mathbb{Z}_p)$

Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\mathbb{C}^n$, $n \ge 2$. Then $\pi \subset S^1$ acts freely on $S^{2n-1}$…
User
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Question 1.1.2 from Hatcher

I'm having trouble understanding this question. We have a path $h$ in $X$ from $x_0$ to $x_1$ and $\overline{h}$ its inverse path. Then a map $\beta _h:\pi_1(X,x_1)\to \pi _1(X,x_0)$ defined by $\beta _h[f]=\left [h\circ f\circ \overline{h}\right…
09867
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Map to $RP^2 \vee S^1$ nullhomotopic

Let $R$ be $S^{1}\vee S^{1}$. Call the first circle by $a$ and the second one by $b$. Let $X$ be space by attaching two $2$-cells to $R$ one via the boundary map $a^{3}$ and the other via the boundary map $ababababab$. Show that any map $f: X\to…
Peyman Morteza
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