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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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Existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure via determinant line bundle

In Dan Freed's notes Exercise 9.30 he outlines the proof of the existence criterion which is that there exists $\tilde{c} \in H^2(M;\mathbb{Z})$ such that $2\tilde{c} = c_1(E)$. His approach is to pass to the determinant line bundle Det$(E) \to M$…
PhysicsMath
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(Non-trivial) local coefficient system which is not a bundle of groups

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. Let me first give the definition of a local coefficient system that I found in [2, p. 257] and [3, p. 35]: Let $X$ be a topological space. A local coefficient system…
Kathy
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Fundamental group of quasi circle(page 79 ,Hatcher) is trivial

I was trying to show that fundamental group of quasi circle(page 79 ,Hatcher) is trivial.I can understand that every loop is precisely zero loop because for any loop if it start at some point it has to come back at that point and there is exactly…
Shivani Sengupta
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how this two space are homotopy equivalent?

May be this is a very silly question but it is somehow not clear to me.... If we take the space sphere with a diameter attached between north pole and south pole then if we start sliding one point towards another then we will get resulting space as …
Shivani Sengupta
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Homotopy equivalence induces an isomorphism

I am having trouble following the argument in Massey for the proof of Theorem 8.3, which states that if $f: X \rightarrow Y$ is a homotopy equivalence, then $f_*: \pi(X, x) \rightarrow \pi(Y, f(x))$ is an isomorphism for any $x \in X$. The proof in…
Bob
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Map of degree two from $S^2$ to the torus $T^2$.

Prove that there is no map of degree two from $S^2$ to the torus $T^2$. I'm struggling with this problem. I've tried lifting the map to the covering space but I'm not sure what to do from there. I keep getting results that I know are wrong. Most…
topman12435
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Show that Borsuk lemma need not hold if $f$ is not injective

The following lemma is called Borsuk lemma which can be found in Munkres' topology (Lemma 62.2). (Borsuk lemma) Let $a$ and $b$ be points of $S^2$. Let $A$ be a compact space, and let $f:A\to S^2\setminus\{a,b\}$ be a continuous injective map. If…
Xiang Yu
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Is there a common way, to find all deck transformations $\Delta(p)$

Suppose $p:E\longrightarrow B$ is a covering projection. I have a general question, on how to find the group of all deck transformations $\Delta(p)$. Is there a common way to do this, or what could be a good approach? Thanks in advance.
Mr.Topology
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Product of coverings is a covering of product space.

I want to prove this statement: Let $p_i: Y_i\to X_i$ with $i=1,2$ be covering spaces. Show, that $p_1\times p_2: Y_1\times Y_2\to X_1\times X_2$ is a covering space. Therefore I want to show, that for every $(x_1, x_2)\in X_1\times X_2$ exists a…
Mr.Topology
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Fundamental Groups of Complements of Knots Algorithm

Is there any clear algorithm to compute the fundamental group of complements of knots?
Sam
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Homotopy type of mapping space

In Ralph Cohen's notes on the topology of fiber bundles (pp.63) he claims that the space of all $G$-equivariant maps from $P$ to $EG$ denoted by Map$^G(P,EG)$ is aspherical, where $EG$ is the total space of the classifying space $BG$ and $P$ is any…
PhysicsMath
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Why does $f(x,z)=(x,z^2/|z|)$ have degree $2$?

Write the $n$-sphere as the set $S^n\approx \{(x,z)\in\Bbb R^{n-1}\times\Bbb C: |x|^2+|z|^2=1\}$, and define a mapping $f: S^n\to S^n$ by $f((x,z)) = (x,\frac{z^2}{|z|})$. Why is $\deg(f)=2$? Background: This is the essential content of a problem…
Eric Stucky
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Resources that explains "Cut and Glue" Technique for Delta Complex?

I am looking for any resources (book/online) that teaches and further elaborates on how the "cut and glue" technique works for $\Delta$-complexes. To be precise, I am looking for techniques and at least some semi-rigorous theoretical framework…
yoyostein
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The space of connections is affine thus contractible?

In Ralph Cohen's notes on the topology of fiber bundles pp.62 he states that, since the space of connections $\mathcal{A}(P)$ (where $P$ is a principal $G$-bundle is affine) it is contractible. I found in Stack Exchange there is a related old…
PhysicsMath
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Graph Homology and Rank-Nullity Theorem

Let $G$ be a connected, directed graph with $v$ vertices and $e$ edges. According to Massey (Ch. VIII, Section 3), the euler characteristic satisfies \begin{align} v - e = \chi(G) = \text{rank} \, H_0(G) - \text{rank} \, H_1(G) = 1 - \text{rank}…
user02138
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