# 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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### $H_q(X;\mathbb{Z})=0$ when X spherical complex with $H_q(X;F)=0$ for all $q>0$ and for all $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$

Suppose that X is a spherical complex with $H_q(X;F)=0$ for all $q>0$ where $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$, as $p$ runs over all primes. I want to prove $H_q(X;\mathbb{Z})=0$. I know ...
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### Why is this not a triangulation of the torus?

I refer to example 4, fig.3.6, p.17 of Munkres' Algebraic Topology. He says the given triangulation scheme "does more than paste opposite edges together". Not clear to me. For those who don't have ...
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### Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of connected/compact/contractible/simply-...
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### covering map $S^n \rightarrow P^n$ is not null homotopic

Here is the problem: Prove that the covering projection $S^n \rightarrow P^n$ is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ...
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### pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
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### Is $S^2/\sim$ a $CW$-complex?

Consider the equivalence relation on $S^2$ define by $x\sim -x$ if $x\in S^{1}$ (we are supposed to see $S^1\hookrightarrow S^2$ as the equator) and $x\sim x$ otherwise. I have some questions ...
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### Decomposition of cohomology group on $S^{n}$

If we have decomposition of cohomology group on $S^{n}$ it looks like $H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}$, where $H^{n}(S^{n})_{\pm}$ cohomology of invariant or anti-invariant $n$ ...
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### Good resource for learning braid theory?

I recently heard about braid theory and read the Wikipedia article on it, and it seems really beautiful. What is a good resource for learning more about it? I have a background in mathematics at the ...
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### Is it true that $X\simeq S^2\vee S^2$?

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim -x$ for every $x$ in the equator $S^1$. Is it true that $X\simeq S^2\vee S^2$, that is, $X$ is homeomorphic to $S^2\vee S^2$?
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### Fibrewise normal, but not functionally normal space

In general topology, if a space is normal, then exist a continuous function which separates two closed sets. This is because on a normal space, you can "put" an open set and it's closure between an ...
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### A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
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### Determining if certian properties of a topological space pass to its image under a quotient map.

A property $P$ of topological spaces is said to "pass to quotients" if whenever $p : X \rightarrow Y$ is a quotient map and $X$ has property $P$ then $Y$ has property $P$. For the following properties ...
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I am currently reading in Hatcher's book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood $N_{\... 1answer 335 views ### Poincare dual of unit circle I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in$ R^2-\{0 \} $is zero, ... 1answer 80 views ### Is$\mathbb{C}P^{\infty}$, the infinite complex projective space simply connected? [closed] Is$\mathbb{C}P^{\infty}$simply connected? 1answer 42 views ### Proving the left lifting property for a map I want to prove the left lifting property for the inclusion map of the sphere into the disk for any fibration$q:X \rightarrow Y$, where$q$is a weak equivalence. I don't know how to draw a square ... 0answers 104 views ### Winding number and homotopy Given two maps$f,g : S^1 \rightarrow S^1$, I want to show that if they have the same winding number, then there is a homotopy between them. Well, we know that we can write them as$f(\exp(2 \pi i t))...
I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
### In the definition of $n$-equivalence, what is the motivation for only requiring surjectivity on the $n$th dimension.
An $n$ equivalence $f\colon X \to Y$ such that the induced map on the homotopy group $f_* \colon\pi_m(X) \to \pi_m(Y)$ is an isomorphism for $m<n$ and an epimorphism for $m=n$. What's the ...