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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What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a vector space is the number of vectors in its basis.…
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Can a division algebra over $\mathbb{R}^3$ be used to construct a counterexample to the hairy ball theorem?

Suppose (for contradiction) that there is a (if necessary associative and/or normed) division algebra over $\mathbb{R}^3$. Is there a simple way to use this to construct a nonvanishing continuous tangent vector field on $\mathbb{S}^2$, and thus…
Student G
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Full (or universal) group C*-algebra of discrete group $\Gamma$

There is a quotation below (C*-Algebras and Finite-Dimensional Approximations): $ \qquad$We denote the $group~ring$ of $\Gamma$ by $\mathbb{C}[\Gamma]$. By definition, it is the set of formal sums $$\sum_{s\in \Gamma}a_{s}s,$$ where only finitely…
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Homotopy type vs. weak homotopy type, and repercussions for EG

This is another basic question. I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point topological space $X$ such that the only…
jdc
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Uniqueness of curve of minimal length in a closed $X\subset \mathbb R^2$

Suppose $X$ is a simply connected closed subset of $\mathbb R^2$. Let $a,b$ belong to $X$. Is it true that there is at most one curve inside $X$ from $a$ to $b$ such that the length of the curve is minimal?
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Homotopy equivalence induces bijection between path components

If $x\in X$, let $C(x)$ the path component of $x$ (the biggest path connected set containing $x$), and similarly if $y\in Y$. Let $C(X)$ and $C(Y)$ the family of all path components of $X$ and $Y$. Let $f:X\to Y$ be a homotopy equivalence. We…
Sylas
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De Rham cohomology of $T^*\mathbb{CP}^n$

I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of $T^*\mathbb{CP}^n$ (seen as a real manifold). Now, this should be equal to the cohomology of $\mathbb{CP}^n$ since the two…
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Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The following are equivalent: $f : X \to Y$ is a homotopy…
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Reference about history of characteristic classes

I'm looking for a good reference about history of characteristic classes. For example, I would like to know how Chern define the Chern class at first in history, or who rewrote the characteristic classes in language of cohomology theory the…
user115322
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Are the fibers of a flat map homotopy equivalent?

At the end of the Wikipedia article on Deformation Retract, there is the following sentence: Two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger space. I was wondering if this has some meaning in…
Brenin
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Visualizing topology of a Vector Bundle

I've started reading Milnor, Stasheff - Characteristic Classes and at page $18$ they proved that $\mathbb{R}^n$-bundle $\xi$ is trivial if and only if $\xi$ admits $n$ cross sections $s_1, \dots , s_n$ which are everywhere independent. Inside the…
Riccardo
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Computation of fundamental group of pseudo circle

A common example of a weak homotopy equivalence which isn't symmetric is the pseudo circle $\mathbb{S}$. Wikipedia gives the following map $f\colon S^{1}\rightarrow\mathbb{S}$ $$f(x,y)=\begin{cases}a\quad x<0\\b\quad…
PeterM
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Homology of stack points

This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the classifying stack $\mathcal{B}G$. This has a (nonrepresentable) morphism to a point:…
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How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand why homogenous equations like $X^2+Y^2=Z^2$ boil…
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Framing Integer associated with a Framed Knot/Link

I have been looking for a clear definition of the n-framing of a knot unsuccessfully. A framing here refers to a choice of homeomorphism between a solid torus neighborhood (a.k.a, tubular neighborhood) of a knot and $S^1 \times D^2$ .I vaguely…
user99680
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