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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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How is this circle in $S^1 \times D^2$ null-homotopic?

This picture is from Hatcher's AT: I have been told that this circle is null-homotopic, but I can't see why. I know $S^1 \times D^2$ is a solid torus, but $A$ is linked with itself. How are we to unwind $A$?
user5826
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The naturality of the cone construction

Let $X$ be a topological space and $CX$ be its corresponding cone. Suppose $g$ is a $n-$ cycle of $X$, I want to show that $g$ is a boundary in $CX$. According to the naturality of the cone construction, I can find a map $Cg: \Delta^{n+1} \to CX$,…
z.z
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Restriction is a homeomorphism between $S^1$

I am doing some problems on Algebraic Topology, this one I came across and currently have no idea how to solve for it, I am glad to have some hints to get me started with (don't give me the full answer, please) if $f : B^2 \to B^2$ is a continuous…
user63279
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How to find the homotopy fiber of bouquet embedding $S^1⋁S^1↪S^1×S^1$

Can somebody please explain how to find the homotopy fiber of bouquet embedding $S^1⋁S^1↪S^1×S^1$
jessicaa
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Fundamental group of Klein Bottle with 2 points removed.

Does anyone know what this group is. Just want to know what the group is. Thanks.
doncarlos
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Let $\gamma\in S_{n+1}$ act naturally on $\Delta^n$, then for some $\sigma:\Delta^n\to X$, $\sigma\circ\gamma=\mathrm{sgn}(\gamma)\sigma$ in homology

Suppose we are given a topological space $X$, and a singular simplex $\sigma:\Delta^n\to X$, where $\Delta^n$ is the convex hull of $e_0,\ldots,e_n\in\mathbb R^{n+1}$. For any permutation $\gamma\in S_{n+1}$, we can define…
Dominic Wynter
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algebraic topology - hatcher 3.3 exercise 17

The following is a question from Hatcher's "Algebraic Topology" “show that homology commutes with direct limits. “ I have tried to solve this problem but I can’t .
user567277
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1 answer

Cell complex structure of real projective plane

A Cell Complex structure of $RP^2$, the real protective plane, is $e^0∪e^1∪e^2$. But I am unable to find this cell complex structure for $RP^2$ from iterative method: What should be the set $X^0$ and then how the set $X^1$ will look like ? Anybody…
Prince Khan
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cohomology of surfaces with local coefficients

Let $X$ be an orientable surface (real manifold of dim 2). Let $F$ be a local system (a locally free sheaf of abelian groups) on $X$. Is is true that $H^n(X,F)=0$ for $n>2$? And that $H^n(X,F)=0$ for $n \geq 2$ if $X$ is non-compact? (Here the…
Visitor
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cell complex structure of circle

I came to know that the cell complex structure of the circle $S^1$ is $e^0∪e^1$. But in my point of view it should be $D^0∪D^1$ as $e^0$ is nothing but an empty set. Can anybody make me understand where am I wrong?
Prince Khan
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Covers of the wedge of three circles

I'm trying to come up with covering spaces of $X= S^1 \lor S^1 \lor S^1$ corresponding to two subgroups of $\pi_1(X)= F_3=\langle a,b,c \rangle$. First, the subgroup $H=\langle a^2, b^2, c^2 \rangle$ and second, the normal subgroup $N= \langle…
user10039910
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Difference between the product group, the direct sum and the free group

The following is from Hatcher's Algebraic Topology: Suppose one is given a collection of groups $G_{\alpha}$ and one wishes to construct a single group containing all these groups as subgroups. One way to do this would be to take the product group…
Koda
  • 1,196
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Proving $\pi_1(X\vee Y, p) = \pi_1(X,p) * \pi_1(Y,p)$

I want to prove the following result: Let $X, Y$ be topological spaces and $X\vee Y$ it's wedge sum on $p$. If there exist simply connected neighborhoods of $p$ on $X$ and $Y$ then $\pi_1(X\vee Y, p) = \pi_1(X,p) * \pi_1(y,p)$ My attempt: If we…
Johanna
  • 405
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fundamental group of quotient space

I found this exercise but I can't do it. the text says: Consider the quotient space $X = T^2/\sim$, where $T^2 = S^1\times S^1$ is the $2$-dimensional torus and this $\sim$ is equivalence relation which identifies two distinct points $p$, $q$ of…
MAX
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degree of a map between Manifold and Hopf theorem

Let $M$, and $N$ be closed orientable manifolds with the same dimensions. Hopf Proved that if $N=\mathbb{S}^n$, then every two continuous maps from $M$ to $N$ with the same degree are homotopy equivalence. In particular, each degree zero map from…
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