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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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Hatcher exercise 2.1.6 (Simplicial homology)

Compute the simplicial homology groups of the $\Delta$-complex obtained from $n+1$ $2$-simplices $\Delta_0^2,...,\Delta_n^2$ by identifying all three edges of $\Delta_0^2$ to a single edge, and for $i>0$ identifying the edges $[v_0,v_1]$ and…
iwriteonbananas
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lifting property of universal cover of $\mathbb{RP}^n$

I have a question. If we have a map $f:\mathbb RP^n\rightarrow \mathbb RP^n$, then can we always lift it to a map $g:S^n\rightarrow S^n$ such that the diagram commutes? $$\begin{array} $S^n & \stackrel{g}{\longrightarrow} & S^n\\ \downarrow{p} & &…
user146507
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Homolgy of the 2d-skeleton of the 4d-cube

I am currently trying to calculate the homology groups for the 2 dimensional skeleton of the 4 dimensional cube. My problem is that I have no idea what this looks like; I know that the 1 skeleton is the hypercube graph, but that's really it. Once I…
TinaBelcher
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Show that if $X$ is a compact metric space and an ANE, such that $H_n (X) \neq 0$, then $X$ cannot be embedded in $\mathbb{R}^n$.

An absolute neighborhood extensor (ANE) is a space $Y$ such that for every metric space $X$, $A$ - a closed subset of $X$, and a map $f:A \to Y$, there exists an open set $U$ containing $A$ such that $f$ can be extended to a map $U \to Y$. $H_n(X)$…
Yoav Bar Sinai
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The definition of a complex oriented cohomology theory

Here are two definitions for complex oriented cohomology theories. A complex oriented cohomology theory $E$ is a multiplicative cohomology theory With a class $x \in \tilde{E}^2(\mathbb{C} P^\infty)$ whose restriction under the composite…
Juan S
  • 10,268
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A question regarding singular homology (The geometrical interpretation of cycles in singular homology)

From Hatcher's Algebraic Topology: Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite $\Delta$-complexes. To see this, note that a singular $n$-chain…
mathamateur
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Prove that ANE space also has HEP

Let $Y$ be a normal space, we say that it is ANE (Absolute Neighborhood Extensor) if for every metric space $X$ and closed subset $A$ of $X$, if $h\colon A\to Y$ then there is an open neighborhood $U$ of $A$ that we can extend $h$ to $h'\colon U\to…
Asaf Karagila
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Why steenrod commute with transgression

I'm reading Hatcher's notes on spectral sequences and he mentions that steenrod squares commute with the coboundary operator for pairs (X,A) which would then explain why these operations commute with the transgression. It says it's because that…
Analyst2
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finding an acyclic but not contractible space

I'm studying singular homology, and have come across the term acyclic. I understood that a contractible space is acyclic, but can't find a counterexample showing the inverse is false. Could anyone suggest me (hopefully the easiest) example?
Keith
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Calculate the de Rham cohomology of the Möbius band

Using Mayer-Vietoris to calculate the de Rham cohomology of the Möbius band $M$, what is the choice of separation? i.e. $M=U\cup V$, which $U$ and $V$ well chosen for calculation?
Ben Li
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Question about the problem that $P^n$ admits a field of tangent $1$-planes if and only if $n$ is odd.

I want to ask the problem 4-C in the characteristic classes written by John W. Milnor. Problem [4-C]. A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a sub-bundle of dimension $k$. Show that $P^n$…
ljh8372
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Euler number of K3 surface

The Euler number of a K3 surface is 24, which can be obtained by using many deep results in algebraic geometry. Is there an elementary way in algebraic topology to get it? For example, let's consider $X=\{z_{0}^{4}+ \cdots +z_{3}^{4}=0\} \subset…
user180625
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Van-Kampen and Covers

Is there a Van-Kampen-style theorem for universal covers? I was looking for a reference. I was looking for something like: given two topological spaces $X,Y$, the universal cover of $X\cup Y$ follows from considering (insert a proper construction)…
Ivan Di Liberti
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What does it mean that the quotient $S^n\to\mathbb{R}P^n$ acts as the identity on the upper hemisphere, and the antipodal map on the lower hemisphere?

I'm not sure how the degree of cellular maps are computed when finding the homology of $\mathbb{R}P^n$. I know $RP^n$ has CW structure with a cell in each degree, and $e^k$ is glued to $RP^{k-1}$ by the attaching map $q:S^{k-1}\to RP^{k-1}$. The…
tegan
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Self-intersections of loops

Let $X$ be a topological space with basepoint $x_0$. Define a map $s\colon\pi_1(X,x_0)\to\mathbb{Z}_{\geq 0}$ by $$s([\gamma])=\min\{\text{number of self-intersections of }\gamma'\colon \gamma'\in [\gamma]\}.$$ Has the map $s$ been studied before?…
Filanto Renika
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