# 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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### $X = S^1 \vee S^1$ so $\pi_1(X) = F\{a,b\}$. Given homo. $\varphi: \pi_1(X) \rightarrow \mathbb{Z}/3$, draw associated cover of $\ker\varphi$.

Let $X=S^1 \vee S^1$ and so that $\pi_1(X)=F\{a,b\}$, the free group on two generators. Let $\varphi:\pi_1(X) \rightarrow \mathbb{Z}/3$ be the homomorphism induced by $\varphi(a)=1$ and $\varphi(b)=0$....
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### Homology groups of orientable surfaces.

Edit: I have a proof here but when I spoke last with my professor, she told me something was close, but not quite. Can someone help me patch this proof? I've been trying to get this down for quite a ...
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### Existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure via determinant line bundle

In Dan Freed's notes Exercise 9.30 he outlines the proof of the existence criterion which is that there exists $\tilde{c} \in H^2(M;\mathbb{Z})$ such that $2\tilde{c} = c_1(E)$. His approach is to ...
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### Universal Cover of wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$.

We are asked to find the universal cover of the wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$. I am second guessing myself on this problem because I ...
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### Does $T^3$ double cover $\mathbb{R}P^3$?

So $T^2$ double covers $S^2$, and $S^2$ double covers $\mathbb{R}P^2$. Therefore, $T^2$ quadruple covers $\mathbb{R}P^2$. I am looking at $\mathbb{R}P^3$ because I am interested in rotations. The ...
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### The circle is not contractible

I know that the circle is not contractible because I know that $\pi_1(S^1)\cong \mathbb Z$. But something is going wrong in my head. Choose a basepoint $*$ on the circle and chose an orientation (...
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### Existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure

In deriving the existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure Ralph Cohen in his notes on the topology of fiber bundle (pp.169) uses the following diagram $\require{AMScd}$ \...
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### Modular curve, $G$-Galois branched cover

I was reading this answer by Pete L. Clark: "For "most" finite simple groups $G$ it is indeed the case that $G=⟨x,y⟩$ where $x$ has order $2$ and $y$ has order $3$. Equivalently, $G$ is a ...
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### (Non-trivial) local coefficient system which is not a bundle of groups

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. For my question the following definition of a local coefficient system is needed ([2, p. 257], [3, p. 35]): Let $X$ ...
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### Counterexample: Homology with Local Coefficients versus Homology with Module Coefficients

Definitions Let $X$ be a nice space with universal covering $\widetilde{X}$. Homology $H_*(X,R)$ with Ring Coefficients $R$ is the homology of \begin{align*} C_n(X;R) :=&\; \text{ free left $R$-...
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### Homotopy: equivalence relation (continuity of homotopy in symmetry)

Let $f_0, f_1: X \rightarrow Y$ be continous on topological spaces $X,Y$. Let $F$ be a homotopy between $f_0, f_1$. Which argument shows that $G(x,t):=F(x,1-t)$ is continous? Seems to be hard to ...
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### Retractions and Isomorphisms of Fundamental Groups

Suppose there is a retraction from $$S^1 \times D^2 \to S^1 \times S^1.$$ Does that then induce an isomorphism $$\pi_1(S^1) \times \pi_1(D^2) \cong \pi_1(S^1) \times \pi_1(S^1)?$$ Which is obviously ...
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### Covering space of wedge of two circles generated by $\{a^{3},b^{2},aba^{-1}b^{-1}\}$

I've been drawing graphs for too long trying to figure it out. I know how to find the generating set using maximal trees, but is there a trick for the reverse problem of given a generating set, come ...
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### Covering Spaces of Wedge of two Circles

I am working on some qual questions, and I don't really comprehend this. Let $X = S^{1} V S^{1}$. The question is asking about covering spaces 7,9, 11, and 12. a. Determine whether or not the ...
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### Any use to thinking of the pullback of a fiber bundle as an integral?

We can think of the pullback of fiber bundle an integral as follows. Suppose $p:E \to B$ is a fiber bundle and $f:A \to B$ is a map. Then we have the pullback bundle $A \times_f E \to A$. The bundle ...
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### Normal covering spaces of wedge of two circles

On page 58 of Hatchers textbook, PDF here: https://www.math.cornell.edu/~hatcher/AT/AT.pdf He says the graphs with "maximal symmetry" are normal covering spaces. This is graphs 1,2,5, 6,7,8 and 11 ...
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### Ham Sandwich Theorem - intuitive proof

Ham Sandwich Theorem. Given 3 measurable "objects" in $\mathbb{R}^3$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single 2-dimensional plane. Can ...
1answer
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### Why are spherical objects so named?

Let $S$ be an object in an abelian category. Then we say S is spherical if $Ext^p(S,S)$ is 0 unless $p = 3$. I know that the cohomology of the three sphere bears some formal resemblence, but it doesn'...
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### Is it possible to define orientability using orientation preserving loops?

Wikipedia says that the orientable double cover corresponds to the subgroup of orientation preserving loops in $\pi_1$ (which is of index 1 or 2 apparently). My questions are: What is an orientation ...
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### Fundamental group obtained by attaching a n-cell with n ≥ 2

I am having trouble with Hatcher's Algebraic Topology P39, Problem 18: Show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^n$ with $n ≥ 2$, then the ...
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### Why is there no retraction between $D^2 \times S^1$ and $S^1\times S^1$?

Why is there no retraction between $D^1 \times S^1$ and $S^1\times S^1$? I have no idea how to prove. I just know that if there is then $\mathbb{Z}\times \mathbb{Z}$ contains a copy of $\mathbb{Z}$. ...
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### Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
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### Is there a contractible space with a free circle action?

Question in title. Seems no to me (some vague intuition here about contracting orbits to a fixed point), but I can't prove it. I'd prefer to be wrong. (I'm curious because I am thinking about group ...
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### On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
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