Mathematics Stack Exchange
Mathematics Stack Exchange

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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Equal winding number implies two paths are path homotopic?

Let $\alpha,\beta:[0,1]\rightarrow\mathbb{C}\setminus\{p\}$ be two (continuous) paths (not necessarily closed) with same endpoints ($\alpha(0)=\beta(0)$, $\alpha(1)=\beta(1)$), we know that if $\alpha\simeq_\mathrm{p}\beta$, then…
Kaa1el
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Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the exercises. I am trying to figure out the (general,…
Haullab
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How to show that the cohomology of a Grassmannian has a basis consisting of the equivalent classes represented by Schubert cycles?

Let $G(r, n)$ be the Grassmannian of the set of all $r$-planes in a $n$-dim vector space. How to show that the cohomology of a Grassmannian has a basis consisting of the equivalent classes represented by Schubert cycles? I am confused since I don't…
LJR
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Mayer-Vietoris Type Sequence For Pushouts

Pushouts in the category $\mathsf{Top}$ of topological spaces exist and under certain conditions are known as adjunction spaces. Rigorously, if is a diagram in $\mathsf{Top}$, then there exists a universal commutative…
Brian Fitzpatrick
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Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the subject, it struck me as the kind of thing for which…
Nick
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(Explicitly) Constructing Deformation Retractions

I'm having trouble building the actual deformation retractions, although I understand the concepts behind them, homotopies, etc. For example, when constructing a deformation retraction for $\mathbb{R}^n-\{0\}$ to $S^{n-1}$, I found that you could…
Dustin Tran
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Algorithm for computing the rank of the fundamental group of a graph?

I've been learning a bit about applications of algebraic topology to graph theory and I'm interested in figuring out how to compute the fundamental group $\pi_1(X,x_0)$ of an arbitrary graph $G=(V,E)$. It seems to me you could use a simple DFS to…
William
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Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
MikeMatsson
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A (somewhat) conceptual proof that the boundary of a fundamental class of a manifold with boundary goes to a fundamental class?

In this set-up, let M be a compact n-dimensional manifold with boundary $\partial M \neq \emptyset$. Assume that M is orientable, and that $[M] \in H_n(M,\partial M;R)$ is the fundamental class of M. If you haven't seen orientability / fundamental…
user101036
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On Frobenius reciprocity theorem

The classical Frobenius reciprocity theorem asserts the following: If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res U})_{H}.$$ The proof in the standard textbook…
Kerry
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$H_1(\mathbb{R}, \mathbb{Q})$ is free abelian

I'm trying to show that $H_1(\mathbb{R}, \mathbb{Q})$ is free abelian, this is another exercise in Hatcher. I'm not sure but I thought I can use the exact sequence $$ \cdots 0 \xrightarrow{f} H_1(\mathbb{R}, \mathbb{Q}) \xrightarrow{g}…
Rudy the Reindeer
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Is it possible to define orientability using orientation preserving loops?

Wikipedia says that the orientable double cover corresponds to the subgroup of orientation preserving loops in $\pi_1$ (which is of index 1 or 2 apparently). My questions are: What is an orientation preserving loop? Is it possible to use it to…
user40167
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Application of Hodge decomposition

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. So how to apply the decomposition to other…
zhangwfjh
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Simply Connected domains.

If $U$ and $U'$ be two domains in $\Bbb C$, and $f$ be a homeomorphism in $U$ and $U'$ then domain $U$ is simply connected $\iff$ $U'$ is simply connected. I found this problem in complex analysis. So I would prefer to know its proof from complex…
Mat He Mat Cian
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Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, but unfortunately I do not…
Seirios
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