Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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Manifolds Resulting from Gluing Tori

I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends ...
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Explanation for a line from a MathOverflow answer

Sometimes I see questions answered on MathOverflow in such a way that I don't really understand the answers. Sometimes I work out what they mean, and other times I can't. I'd like to ask for more ...
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if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,show that $p^{-1}(A)$ is path connected.

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,also $A\subset X$ is a path connected subset,show that $p^{-1}(A)$ is path connected. I suppose that ...
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Hyperbolic surface

Let $X=S_1\times S_1−Δ$ , where $Δ=\{(x,y)∈S_1\times S_1|x=y\}$. I know that this is (one of the) usual model for the cylinder , but how to proof that it is a hyperbolic surface? Any help is ...
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Connectivity and Euler characteristic for surfaces

I learn the concept of connectivity from Hilbert's Geometry and the Imagination as follows: A polyhedron is said to have connectivity $h$ (or to be $h$-tuply connected) if $h-1$, but not $h$, ...
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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 ...
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Do the composition of two Knots always yield a distinct knot (ignoring orientation)?

I would greatly appreciate if I could get some help in clarifying my understanding. (This is a special topic I am studying as a 2nd year University student - I haven't taken topology yet - so please ...
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References Request (Poincare Duality via Unimodular pairing, de Rham isomorphism)

Chapter 0 of Griffith's Harris contains a substantial amount of topological results that I have not seen elsewhere, namely: Poincare duality as a unimodular intersection pairing on homology. Also ...
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3-fold connected covers of punctured torus

I am interested in finding all the connected surfaces (up to homeomorphism) that can be described as $3$-fold covering spaces of the torus with a disc deleted (in both cases that the disc is closed or ...
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Isomorphic fundamental groups result in homeomorphism?

I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic?
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Why are there cubics in a Calabi-Yau manifold?

I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.) ...
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Why is $\pi_1(X,x_0)$ a group?

I want to show that $\pi_1(X,x_0)$ is a group. I am told that $e(t) := x_0$ is the identity element. Now, I am struggling to show that it is an identity element, and also that the inverse of an ...
$H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$
I can't seem to figure out why $H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$. In my notes it says: ''...Since $\varepsilon \partial_1 = 0$, $\varepsilon$ vanishes on $im \partial_1$ and hence ...