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.
22,737 questions
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Is there a non-Serre-fibration that has unique path lifts?
Are there topological spaces $X, X'$ and a map $p: X' \to X$ such that:
For every path $\alpha: I \to X$ and $\hat{\alpha}_0 \in X'$ such that $p(\hat{\alpha}_0) = \alpha(0)$ there exists a unique ...
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Which compact Lie groups have only finitely many non-zero homotopy groups?
Is it possible to determine which compact Lie groups $G$ have only finitely many non-zero homotopy groups; that is for some positive integer $N$ we have $\pi_n (G) = 0$ for all $n \ge N$? I have no ...
- 4,195
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Fibre bundles with spheres as total space, base space and fiber.
Recall Adams' theorem, which states that a map $f:S^{2n-1}\longrightarrow S^n$ has Hopf invariant $\pm 1$ only if $n=2,4,8$.
Now, the Wikipedia page on the Hopf fibration states that the only fiber ...
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What is $\underset{n}{\operatorname{colim}} \tilde{C}_\ast(K(\mathbb{Z}, n)) [-n]$?
Let $K(\mathbb{Z}, n)$ be the Eilenberg-MacLane space of $\mathbb{Z}$ in dimension $n$. Let $\tilde{C}_\ast(K(\mathbb{Z}, n))$ be the augmented chain complex of $K(\mathbb{Z}, n)$, i.e.
$$
\tilde{C}_\...
- 5,583
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Finite-Sheeted Covering Spaces of $\Sigma_{3}$
I am trying to figure out the list of all possible $k$-sheeted covering spaces of the surface of genus 3, $\Sigma_{3}$, where $k \geq 1$ is some positive, finite integer. So far, I have managed to ...
- 81
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Homotopy type of the complement of a divisor with normal crossing in a compact manifold
Let $M$ be a compact smooth complex manifold and $D$ a divisor with normal crossings in $M$. There is a theorem that $M$ is homotopy equivalent to a finite CW complex. Is $M-D$ homotopy equivalent to ...
- 585
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Cyclic Space Filling Formula [closed]
$$C = \operatorname{rank} H^k(X)$$ $$a_1 = C/x+y+z+t+\ldots$$ $$a_V = |C/x-y-z-t-\ldots|$$ $$P(x)' = \sum_{i=1}^{n} ia_i x^{i-1}$$ $$i=1$$ $$a_i = a_1, a_2, a_3, a_4, \ldots$$ $$a_1 \leq a_i \leq a_V$$...
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What's important about holomorphic covering maps in Riemann's Uniformization Theorem?
I know the following statement of the Riemann mapping theorem:
Let $\Omega \subsetneq \mathbb C$ be a simply connected domain. Then $\Omega$ is biholomorphic to the unit disk $\mathbb D$.
I ...
- 3,031
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Is the fundamental group of this quotient space of the cylinder $\mathbb{Z}$?
We have to calculate the fundamental group of this quotient space of $S^1 \times I$:
We thought we could retract by deformation the two circumferences into $S^1$ (identifying the vertices and edges) ...
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Generalizations of the Gauss-Bonnet theorem
I know about the Gauss-Bonnet theorem, and I know about one of its generalizations (the Chern-Gauss-Bonnet theorem), but the former is about the Gaussian curvature, while the other is about the ...
- 221
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Connectedness of fiber products of the form $X \times_Y X \times_Y \cdots \times_Y X$
Let $X$ and $Y$ be connected topological spaces with two covering maps $f$ and $g: X \to Y$. Let
$$X^{(n)} = \{ (x_1, x_2, \dots, x_n) \in X^n : g(x_i) = f(x_{i+1}) \ \text{for}\ 1 \leq i \leq n-1 \}.$...
- 65.6k
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Deformation Complexity Formula
$$\digamma = \int_{Ric(g)}^{n} \frac{\frac{\partial X}{\partial Y} t_0}{\tau_{Zar}} \cdot T_0 \, d\kappa$$
($n$ is the dimension in which the object is located, $T_0$ is the Kolgomorov space (the ...
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Prove a closed disc cannot intersect a 3-leaf fold
This is a follow-up from this question that turned out to have a negative answer.
Suppose we have a space homeomorphic to an open book with a single page.
We have a subspace $D$, homeomorphic to a ...
- 12.1k
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How many closed 2-cells are needed to cover THIS CW complex?
Consider the pictured hollow surface. Take a hollow cylinder (with top and bottom discs) and place the base on a flat disc, so the the bounding circles meet at a single point. Then draw a vertical ...
- 12.1k
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Fundamental group of a topological manifold
If we assume that a topological manifold is a topological space locally Euclidean, Hausdorff and second countable we can proof that the fundamental group (at any fixed point) is coutable (for example ...
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