# 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.

13,523 questions
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### Local systems as solution of PDE

I want to understand the correspondence between locally constant sheaves and vector bundles with flat connection on a manifold $X$. Given a local system $\mathcal L$, it is clear how to define a ...
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### quotient map of locally compact spaces [on hold]

Suppose $X,Y$ are non-compact locally comapct Hausdorff spaces,suppose we have a quotient map $Q:X\to Y$,can we conclude that $Q$ is proper?
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### Fulton, Example 3.2.8 [on hold]

Example 3.2.8 Let $X$ and $Y$ schemes, $p$ and $q$ the projections from $X \times Y$ to $X$ and $Y$, $E$ and $F$ vector bundles on $X$ and $Y$, $\alpha \in A_{*}\alpha$, $\beta \in A_{*}Y$. Then \...
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### Hatcher's proof of proposition (c) for covering spaces

I am currently going through the proof of Hatcher's Algebraic Topology, and I am having some difficulty regarding one of his assertions in the section of the proof given below: I am unable, in ...
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### What is the $S^1$-equivariant cup product on $S^2$?

Consider the sphere $S^2 = \mathbb{CP}^1$ with the $S^1 = \{ \tau \in \mathbb{C} \mid |\tau| = 1 \}$ action given by $$\tau \cdot [z_1, z_2] = [\tau ^ k \cdot z_1, z_2]$$ The corresponding $S^1$-...
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### Relationship between homotopy pushout and ordinary pushout

I'm trying to understand the homotopy pushouts and currently looking at the homotopy cofiber. For two maps $f \colon C \to A$ and $g \colon C \to B$ we defined the homotopy pushout to be the regular ...
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### Why $f^{'}_{*}$ group homomorphism exist in 'only if' part of the lifting criterion proof?

In Hatcher, the lifting criterion states (Prop 1.33): Suppose given a covering space $p: (X^{'},x^{'}) \rightarrow (X,x_0)$ and a map $f: (Y,y_0) \rightarrow (X,x_0)$ with $Y$ path-connected and ...
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### calculating the De Rham cohomology of $\mathbb RP^2$ using Mayer Vietoris

View $\mathbb RP^2$ as $\mathbb RP^1$ attached with a 2-dimensional open unit ball $e^2$. Let $U$ be an open neighborhood of $\mathbb RP^1$ and $V$ be $e^2$. Then $U \cap V$ deformation retracts onto ...
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### Proof of existence of spectral sequence.

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $\tilde{\Phi_*}$ in the diagram it's written that: ...
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### Thom class in homology for defining orientability of vector bundles

A rank $r$ real vector bundle $p : E \to B$ is said to be orientable if there is a Thom class $\tau \in H^r (D(E),\partial D(E) ; \mathbb{F})$ (where $D(E)$ is a unit disk bundle and $\mathbb{F}$ a ...
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### Confusion about topology on CW complex: weak or final?

The topology of the CW complex is defined to be the weak topology: given the sequence of inclusions of the skeleta $X_0 \subseteq X_1 \subseteq_ \cdots$ a subset $A \subseteq X = \cup X_i$ is open iff ...
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### Corestricting a weak homotopy equivalance

Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$ be a continuous map. Recall that $f$ is a weak homotopy equivalence iff $f$ induces group isomorphisms on the homotopy groups, i.e.: \...
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### Prove that there is no retraction from $X$ to the image of a loop $\gamma(t)$

Let $X = S^1 \times D^2$, pick $x_0 \in X$. Suppose $\gamma$ is an embedded loop based at $x_0$ which represents an $n \in \mathbb{Z} = \pi_1(X)$ with $n \neq \pm1$. Prove that there is no retraction ...
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### Crushing a central dividing circle in $\Sigma_2$ to a point

Crushing a central separating circle in $\Sigma_2$ to a point So presentation of $\Sigma_2$ is $<x_1,y_1,x_2,y_2:[x_1,y_1][x_2,y_2]>$. I understand that crushing this separating circle to a ...
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### Reference for homology of real projective space in a field.

I need a reference in a book for the computation of the homology of real projective space with coefficients in an arbitrary field. I do know how to do the computation, and I also found an online ...
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### ext sheaf and cohomology

Let $\mathcal{F}$ be a sheaf on $\mathbb{P}^{3}$ with $\mbox{dim}(\mathcal{F}) = 0$. It's true that cohomology $H^{i}(\mathbb{P}^{3}, ext^{3}(\mathcal{F}, \mathcal{O}_{\mathbb{P}^{3}})) = 0$ for \$i = ...