# 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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### The Incluson Map $S_1\to S_1\times S_1$ Induces an Injection in First Homology.

Let $T=S^1\times S^1$ be the torus and $A=S^1\times \{x_0\}$ be the "vertical" circle in the usual depiction of the torus as a tyre tube sitting "horizontally". Let $i:A\to T$ be the inclusion map....
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### Proving weak homotopy equivalence of a map

This is a modification of the question I previously asked here. Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting ...
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### What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...
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### Homology Groups of Tangent 2-Spheres

I have been trying to compute the Homology Groups $H_n$ of two tangent 2-Spheres (we will call this space, X). By previous results, I am able to easily determine that $H_0(X)$ is isomorphic to the ...
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### Cayley graph of the fundamental group of the 2-torus

Does anybody knows some description or some good picture on the internet for the Cayley graph of the fundamental group of the 2-torus, when this is constructed by connected sum of two torus. I know, ...
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### The Euler Characteristic of $\mathbf RP^2$ is a Fraction.

Problem 22 in Section 2.2 in Hatcher's Algebraic Topology reads For $X$ a finite CW complex and $p:\tilde X\to X$ an $n$-sheeted covering space, show that $\chi(\tilde X)=n\chi(X)$. Here $\chi$ ...
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### Non contractible space with contractible suspension

I know that there exist non-contractible spaces $X \not\simeq \ast$ with contractible suspension $\Sigma X \simeq \ast$. For instance the 2-skeleton of the Poincaré homology 3-sphere is such a space. ...
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### Correspondence $\{$principal $G$-bundles on $M\}\leftrightarrow\{$conjugacy classes of homomorphisms $\pi_1(M)\to G\}$

Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ...
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### In the quotient topology $D^2/{S^1} \cong S^2$

Let $X = D^2 = \{(x, y) \in R^2\ : x^2 + y^2 \le 1\}$ be the closed unit disc (in the standard topology). Identify $S^1$ with the boundary of $D^2$. Now I have to prove that $$D^2/S^1\cong S^2$$ I ...
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### Characterizing spaces with no nontrivial covers

I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
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### Hatcher basic terminology/phrasing

I'm trying to self-study some algebraic topology, reading Hatcher. His questions seem much less straightforwardly worded than Munkres - with Munkres it was always clear that you weren't expected to ...
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### Proof of More Flexible Mayer-Vietoris for Calculating Homology Groups

On pg. 150 of Hatcher's Algebraic Topology, the author writes: Let $X$ be a topological space and $A$ and $B$ be subspaces of $X$ such that $X=A\cup B$. Suppose there are open subspaces $U$ and $V$ ...
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### Preimage of a simply closed curve under the two-dimensional antipodal map

Suppose $p:S^2\to P^2$ is the quotient antipodal map, and $J$ is a simply closed curve in $P^2$, then $p^{-1}(J)$ is either a simply closed curve in $S^2$, or two disjoint simply closed curves in $S^2$...
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### Cohomology class of zero set of a section

Say we have a rank $r$ smooth vector bundle $E\to X$ and a smooth section $s:X\to E$ of it. Shortly after the 30 min mark in this video Joe Harris defines the $r$th Chern class of this bundle to be ...
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### Relative Homology of the Mapping Cylinder w.r.t a Subspace

Given a continuous map $f:X\to Y$, the mapping cylinder of $f$ is defined as the space obtained from $(X\times I)\sqcup Y$ by identifying $(x, 1)$ with $f(x)$ for all $x$. Let $f:S^n\to S^n$ be a ...
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### Bott's topological obstruction to integrability

Let $M$ be a smooth manifold, $E\subseteq TM$ be an integrable distribution and $\pi:TM\to Q=\frac{TM}{E}$ the quotient bundle with $\dim Q=q$. Bott showed that Pont^k(Q)=0 \qquad \...
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### Proving that a map is a weak homotopy equivalence

Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting diagrams where all the morphisms are serre fibrations \begin{...
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### How to know if you are “tough enough” to study Algebraic Topology [closed]

I am graduating with a BA this summer and I am very interested in topology. I admit it, I never went that deep into topology and all I know is about point-set topology (metric spaces etc.) but from ...
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### Why does Van Kampen Theorem fail for the Hawaiian earring space?

The Hawaiian earring space has notoriously complicated fundamental group, and is essentially not as simple as the wedge sum of countably many circles whose fundamental group is straightforwardly given ...
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### Degree formula for smash product

Let $f : S^n \to S^n$ and $g : S^m \to S^m$ be two maps with degrees $d_f$ and $d_g$ respectively. These two map gives rise to a map $f \wedge g : S^{n+m} \to S^{n+m}$. My question is how the degree ...
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### submanifold with same homology

Suppose $M$ is a manifold without boundary, and $N\subseteq M$ is any submanifold, possibly with boundary. If $H_*(N)\cong H_*(M)$, is it necessarily true that $N\cong M$?
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### Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration). I know that $S^3$ is simply ...
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For every continuous function $f: S^1\to S^1$ with $f(1)=1$ we define the "function degree" $deg(f)$ as winding number of the path $f\circ w_1$ Show, that: a) $deg(f)=n\Leftrightarrow f\circ ... 0answers 35 views ### Proving that Emb($D^m,N$) is homotopy equivalent to$V_m(TN)$I am reading online lecture notes by John Francis on h-principle. I want to prove that Emb($D^m,N$) is homotopy equivalent to$V_m(TN)$where$V_m(TN)$is stiefel bundle of the tangent bundle on$N$.... 1answer 50 views ###$\pi_1(S^n) = 0$for$n \geq 2$I have few of questions for the following proof in hatcher's book. (1)Why f being continous imply that for each$s \in I$has an open neighborhood$V_s$in I mapped by f to some$A_{\alpha}$. (2)Why ... 0answers 63 views ### Generalized Euler Characteristic I was asking myself what kind of generalizations of the euler characteristic are there? I've heard about the homotopy cardinality, but I was rather interested in a construction involving generalized ... 0answers 84 views ### Topology on$\mathcal{C}(X,Y)$to work with homotopy. We know that the compact open topology on$\mathcal{C}(X,Y)$is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ... 1answer 47 views ### Cohomological AHSS for projective space$\mathbb{C}P^n$ALWAYS collapses at the second page While I was computing the cohomology ring$E^*(\mathbb{C}P^n)$for$E$an oriented ring spectrum, via AHSS I realised that orientation is not a necessary hypothesis in order to compute the cohomology ... 1answer 50 views ### Is the rank of this group always finite? Why? Take$U$an open subset of the plane. Consider$C_0(U)$the free abelian group over the points of$U$,$C_1(U)$the free abelian group over continuous paths (i.e. continuous maps$[0,1]\to U$), and$...
Suppose $X$ is a manifold of dimension $n$ and $f:Y \to Z$ is an $n-$connected map. Then I want to show that given any map from $g:X \to Y$, the composite map $f \circ g$ is nullhomotopic. Definition ...