# 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.

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### Lift of a continuous map $f:Y\to X$ to a covering space $Z$ given that loops in $Y$ lift to loops in $Z$

Suppose $X, Y$, and $Z$ are topological spaces. If necessary, you may assume that they are nice (manifolds). I am looking for the following result to be true even without this assumption however, so ...
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### Polyhedral complex without ambient space

I would like to know the most commonly used terminology for the following simple object in combinatorics/topology. I restrict to the two dimensional case since this is what I am mainly interested in: ...
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### Intersection Number of Stable and Unstable Manifold for Hyperbolic Fixed Point of ODE?

Let $Q^n$ be a closed, Riemannian manifold, $TQ$ its tangent bundle with the canonical lift of the Riemannian metric (as outlined in do Carmo) and the resulting compatible triple, $\xi$ a complete ...
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### Homology is homotopy invariant, fake proof with simplicial methods?

Here is a maybe false proof that I came up with that homology of topological spaces is homotopy invariant. I'm thinking that it is indeed fake because why hasn't anyone else come up with this much ...
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### Beginner Questions on CW-Complexes

As a beginner, I am struggling a bit with CW-complexes. I'm reading Hatcher, chapter 0. So I want to pose a few questions that are almost embarrassing to me but I believe it is important to ask such ...
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### If a linear transformation maps each basis vector $e$ into the complement of the span of $e$, then its matrix has diagonal entries all zero

I'm currently reading the proof for Lefschetz fixed point theorem in Hatcher's Algebraic topology page 179. The conclusion in the last lines of the proof relies on the fact that If a linear ...
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### Applications of higher topos in geometry and topology

Higher topos and derived algebraic geometry are relatively new areas and probably there are fewer people working on them compared to the majority of topologists or geometers. I haven't found geometers ...
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### Spectral sequence with field coefficients

In the situation of the Serre spectral sequence for a fibration $F \rightarrow E \rightarrow B$, when can I say that the cohomology of $E$ with coefficients in a field is the direct sum of the ...
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### Why is the group of units in a valued ring a topological group?

Let $A$ be a ring and $v$ be a valuation on $A$ with value group $\Gamma$. The open sets are: $U=\{a \in A:v(a)< \gamma\}$ for $\gamma \in \Gamma$. How should I show that the group of units ...
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### Bounding $2$-simplexes in punctured plane in $\Bbb R^2 - \{0\}$ intuition

From Rotman's Algebraic Topology concerning developing the intuition on homology functors: The question we ask is whether a union of $n$-smplexes in $X$ actually is such a boundary. Consider the ...
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### Genus and arithmetic genus?

In the Wikipedia of Arithmetic genus, for a complex projective manifold of complex dimension 1, the arithmetic genus satisfies $\chi=1-g$. I also see the relation $\chi=2-2g$ for a connected, ...
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### Formula for juxtaposition of paths defined by equivalence classes?

I am interested in the equations for the juxtaposition of two paths $\alpha$ and $\beta$. I have the following equivalence relation; \begin{equation} (x,y) \sim (x',y') \iff x-x', q-q' \in \{0,1\} \...
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### Is a solid sphere with a “bubble” in the middle topologically the same as a torus?

Suppose there was a solid sphere or ball with radius 2 such that a sphere with radius 1 was removed from the center making a hollow cavity. A sphere with a "bubble in it" if that's easier to visualize....
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### Proving continuity for composed homotopy

Suppose $F$ and $G$ are homotopies between $f:X\rightarrow Y$ and $g:Y\rightarrow Z$, respectively. How to conveniently show that $$H(x,t)=G(F(x,t),t)$$ is continuous? I know how to do this recursing ...
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### If $A/B$ is homeomorphic to $C/D$, where $B\subset A$ and $D\subset C$, then is $H_i(A,B)=H_i(C,D)$?

On pg 125, in his book "Algebraic Topology" Hatcher makes a claim that is analogous to the following: If $A/B$ is homeomorphic to $C/D$, where $B\subset A$ and $D\subset C$, then $H_i(A,B)=H_i(C,D)$. ...
Let $X\cup_f Y$ be an adjunction space and $q:X\coprod Y\rightarrow X\cup_f Y$ be the associated quotient map. Let $A\subseteq Y$ be closed and $f:A\rightarrow X$ be a continuous map (this is called ...