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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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21356 questions
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Hatcher Algebraic Topology P428 Exercise 2: Fibering of spheres has hopf invariant 1 or -1.

I am trying to finish an exercise on Hatcher's Algebraic Topology, it is at page 428, stated as follows: Show that if $S^k\to S^m\to S^n$ is a fiber bundle, then $m = 2n-1, k =n-1$ and if n > 1 the map $S^m\to S^n$ has Hopf Invariant $\pm 1$. I have…
北方火柴
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Pointed and Free homotopies $Y\rightarrow K(\pi,1)$

Let $(Y,y)$ be a CW complex (without restrictions) and $K=K(\pi,1)$ an Eilenberg-MacLane space for the abelian group $\pi$ and $*\in K$ a chosen base point. In particular, $\pi_1(K,*)\simeq\pi$ is abelian, therefore $\pi_1(K,*)\simeq H_1(K,\mathbb…
Alessandro
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Let the topological space $X_n$ be obtained from $S^n$ by identifying three distinct points. Find the fundamental group of $X_n$.

Let the topological space $X_n$ be obtained from $S^n$ by identifying three distinct points, i.e. $X_n = S^n/\{p, q, r\}.$ Find the fundamental group of $X_n$.
dunkindonuts
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A little help on the homology of a torus relative to a circle

First I'll go through my working. Throughout we assume the homology groups of the torus and circle are known. Let $X=S^1 \times S^1$ be the torus, and $A=S^1 \times \{1\}$. The following is part of a long exact sequence: $$ \dots \to H_n(X) \to…
Sputnik
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Is the interior of a closed, contractible set contractible if it is path connected?

My question is closely related to my other question. Since I got the answer for what I asked I’ve decided to ask my modified question separately. If $A$ is a closed, contractible subset of a topological space $X$ and if $A^\circ$ is path connected…
R_D
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Is the "good pair" relation transitive?

For $X$ a topological space and $A$ a subspace, we say that $(X,A)$ is a good pair if $A$ is closed and there is a neighborhood of $A$ which deformation retracts to $A$. Let $X$ be a topological space, and $B \subseteq A \subseteq X$ be two…
QuinnLesquimau
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Local degree of $z\mapsto z^n$

I'm currently computing the local degree of $f:S^1\to S^1$ by $z\mapsto z^n$ with $n>0$. To do this, I tried to compute the local degree $\deg f|x_i$ where $x_i\in f^{-1}(1)$. Since $f$ is a local homeomorphism, $\deg f|x_i =\pm 1$ for each $i$. But…
one potato two potato
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Prove that map from $X$ to $RP^2\vee S^1$ is null homotopic

I am doing this UW Madison geometry qualifying exam, and I got stuck on this one : Let $R=S^1\vee S^1$ as in the figure, and $X$ be the cellular 2-complex obtained by attaching two 2-cells to $R$ as indicated. Let $Y=RP^2\vee S^1,$ show that every…
Simplyorange
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The Lefschetz Number: A Geometrical Interpretation

Well, my question is rather brief and straightforward: does anyone know of any good geometrical interpretation of the Lefschetz number, specifically, if $f$ is a continuous map $X \rightarrow X$, $X$ being a compact triangulable space, then the…
StormyTeacup
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Why defining homotopy on functions instead of spaces?

Apparently, the definition of homotopy formalizes the idea of continuous transformation between two things. (*) Let's take this motivating example: I have $ S,S' \subset \mathbb{R}^3 $ two surfaces in space. If I wanted to define a continuous…
rod
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Homeomorphisms between lens spaces and Poincaré conjecture

It is known that $L(p,q)\cong L(p,q')$ if and only if $q\equiv \pm (q')^{\pm 1}\mod p$ where $$ L(p,q)\cong{\mathbb{S}^3}/\mathord{\sim} $$ is the quotient space generated by the $\mathbb{Z}_p$-action $$ \rho(z_1,z_2)=(\zeta z_1,\zeta^q…
Aner
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Induced map in (co)homology in a map from torus to sphere

Let $f:T^2 \to S^2$ be the map that collapses the 1-skeleton $S^1 \vee S^1$ Compute $f_∗$,$f^*$. So start with the usual CW strucutre on the sphere (1 0-cell and 1 2-cell) and the torus (1 0-cell, 2 1-cells and 1 2-cell). Now all the boundary maps…
Juan S
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Action of fundamental group on n-th homotopy groups for RP^n

Is there any short way to see that the action of $\pi_{1}(RP^{n})$ on $\pi_{n}(RP^{n}) = \mathbb{Z}$ is trivial for $n$ odd and nontrivial for $n$ even? Maybe something without much machinery (smth related with orientability perhaps?) Thank youuu!
Tikks
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Find all covering spaces of Torus $S^1 \times S^1$ up to isomorphism.

The fundamental group of the Torus is $\mathbb{Z} \times \mathbb{Z}$, and unless I'm wrong all the subgroups are one of the following forms: (Grids) $m\mathbb{Z} \times n\mathbb{Z}$ for $m, n \in \mathbb{Z}$ (Lines) $\mathbb{Z} \times (m\mathbb{Z} +…
Rotatopotato
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$\pi_1(X)$ finite, show $f:X \to S^1$ is nullhomotopic

Suppose that $\pi_1(X)$ is a finite group. Show that any map $f:X \to S^1$ is nullhomotopic. My attempt: Since $\pi_1(X)$ is finite and $\pi_1(S^1)=\mathbb{Z}$ torsion-free, then the induced homomorphism $f_*: \pi_1(X) \to \pi_1(S^1)$ has to be…
Dávid Natingga
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