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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Does group of deck tranformations acts transitively on each fibre if it acts traansitively on one fiber?

i am reading bredon "Topology and Geometry " It states that if we have a covering map p : X ->Y s.t. p(x) = y.X,Y are Hausdorff, path connected and locally path connected etc. I have 2 questions: It states that There exist $\alpha \in N(p^*(\pi…
rohit
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A question about the definition of the Tor functor

On pg 278 of Bredon's "Topology and Geometry" says the following" Let $0\to A'\to A\to A''\to 0$ be a short exact sequence of abelian groups, and let $M$ be another abelian group. Then the following long exact sequence is induced: $0\to…
user67803
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A long exact sequence of free Abelian group is the direct sum of very short exact sequences.

A long exact sequence of free Abelian group is the direct sum of very short exact sequences. The definition of short exact sequences doesn't seem to be very common from what I can see online: An exact sequence is very short if it has at most 2…
Haikal Yeo
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All maps from a CW complex to S^1 null-homotopic implies finite first homology

So given a connected compact CW-complex $X$, a quick covering space argument shows that if $H_1(X)$ is finite, then every map $X \to S^1$ is null-homotopic. I was curious if the converse was true: given such a CW-complex $X$, if every map $X \to…
squiggles
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Intuition behind Monodromy Action

I am learning a topological concept called monodromy action that I am having difficulty with, may I ask what is the intuition behind this concept and how does it relates to the lifting property? In particular, if I identify the torus with $S^{1}…
Sophie
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Correspondence between $[X;K(G,n)]$ and $H^n(X;G)$.

I learnt this from Spanier and it is not very clear to me geometrically... If I take a cohomology class in $H^n(X;G)$, is it possible for me to get an idea what exactly this map is in $[X;K(G,n)]$? For example, how can I possibly compute the…
ah--
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Chain map of isomorphic chain complexes

If we know that the homology groups of two chain complex are isomorphic to each other, can we say that there is a chain map between these two chain complexes？ If so, how can we define the chain map?
user53800
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Fundamental group of two 2-spheres attached at a point

Just wanted to confirm if my proof is correct and complete, trying to learn Van-Kampen Theorem. Question: Find the fundamental group of two copies of $S^2$ attached at a point . Proof: We claim that $\pi_1(X)$ is trivial. Let the two copies of…
nonlinearism
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Lovers and haters via the Jordan curve theorem

In J. Roe's Winding Around and V. Arnold's Ordinary Differential Equations there is a beautiful story attributed to N. Konstantinov: A pair of lovers travels from city $A$ to city $B$ via two different roads. If the distance between them is at least…
Paweł Czyż
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Applications of universal coefficient theorem

I'm not really sure what I want to ask here, which isn't a great start for a question, but nonetheless... I am wondering if there are some nice results that we can get from considering (co)homology with coefficients in an arbitrary abelian group.…
Juan S
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To show that Warsaw circle is simply connected

I am doing exercise from algebraic topology book which says show that fundamental group of Warsaw circle is trivial, where Warsaw circle is C=$A_1\cup A_2\cup A_3 \cup A_4 $, where $A_1=\{(x,\sin\frac{\pi}{x}):0
ogirkar
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Weak deformation retraction

This is part of a problem from Hatcher's book. Let $Y$ be the subspace of $\mathbb{R}^{2}$ shown in the picture. Let $Z$ be the zigzag space of $Y$ indicated by the heavier line. Show that there is a deformation in the weak sense (i.e. a homotopy…
cyc
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Bing's House and homotopies

Muestra que si f es la función que encaja a $S^{1}$ en la circunferencia que rodea al cilindro mayor, por la mitad, de la casa de Bing, entonces f es homotópica a una constante. Show that if $\,f\,$ is the function embedding $\,S^1\,$ in the…
user71495
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Algebraic Topology Disjoint Union, Lee

As this is my first post, I hope that what I write is pretty clear and isn't too disappointing. Without further ado, I am quite curious with regards to my understanding of Disjoint Union Spaces. (I am currently self studying from Introduction to…
Moe Khaled Bin-Lateef
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complex of abelian groups is split

Let us call a complex of abelian groups $C_* = \{C_n\}$ split if it admits a direct sum decomposition $$C_* = H_* \oplus B_* \oplus D_*$$ where the differential $d = \{d_n : C_n \to C_{n-1}\}$ in $C_*$ vanishes on $H_*$ and $B_*$ and maps $D_*$…
Koam
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