# 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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### Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
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### “Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
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### An example of infinite Betti number refers to G-covering, and the motivation of G-covering

I saw from a book that if G is infinite, then when considering G-covering: $p:X'\to X$, the P-th Betti number of $X'$ can be infinite. Can you give me an example of that. Also I am curious why we need ...
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### Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
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### Reference for: simple closed curves generate the fundamental group

In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not self-...
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### Homotopy of two circles contained in an open ball.

The following question is on my homework assignment and I have no idea how to even start answering it: Are any two distinct $S^1$ → $B(0,r)$ maps Homotopic? You can assume the circles are simple-...
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### Representing Covering Spaces by PErmutations

I am having trouble understanding the exposition in the subsection titled Representing Covering Spaces by Permutations in Section 1.3 of the book Algebraic Topology by Hatcher. Hatcher starts by ...
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### adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
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### Fundamental group of simple graphs

Find the fundamental groups of graphs A, B and C as shown: They look simple but I am unsure what their fundamental groups would be. I was thinking that $A$ and $B$ are generated by just one ...
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### The space of connections is affine thus contractible?

In Ralph Cohen's notes on the topology of fiber bundles pp.62 he states that, since the space of connections $\mathcal{A}(P)$ (where $P$ is a principal $G$-bundle is affine) it is contractible. I ...
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### How does a boundary operator act on a 2-simplex?

Let $A$ be a 2-simplex with vertices $\{0, 1, 2\}$. I want to show that $\rho_1\circ\rho_2 (A)=0$, where $\rho$ is the boundary operator. How do I go about doing that? The major problem that I ...
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### Nonzero-homologous simple loop in Mobius band only winds once

I have a question as follows: Let $C$ be a closed curve in the Mobius band without self intersections. Prove that if $C$ is of non-zero homology, i.e., $C$ does not bound any face, then $C$ winds only ...
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### Can the K-theory of a space be a field?

If $X$ is a compact Hausdorff topological space, is it possible to $K(X)$ be a field considering the operations over vector bundles, $\oplus$ and $\otimes$? It is known that $K(X)$ has a ring ...
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### Barycentric subdivision proof

In the proof of barycentric subdivision in singular homology, we take the subdivision operator $b: C_q(X) \to C_q(X)$, and do some algebra to define an operator $b^\infty$, which applies $b$ "as many ...
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### Do smooth compact connected manifolds admit CW compositions with a single 0-cell?

I have stated this question incorrectly before, sorry.
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### Is a simplex with permuted vertices $\pm$homologous to the original?

Take a singular $n$-simplex $\sigma: \Delta^n \to X$, where $\Delta^n\subset \mathbb{R}^{n+1}$ is the convex hull of the standard basis, with the obvious vertex ordering. Then one can obtain $(n+1)!$ ...
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### Is there a projective morphism from the quadric surface to the projective plane with degree 1?

Is there a projective morphism from the quadric surface $\mathbb{P}^1\times\mathbb{P}^1$ to the projective plane $\mathbb{P}^2$, with degree $1$?
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### The universal property of pullback bundle

The universal property says the following: for any pair of maps $i: Z \to X$ and $j: Z \to E$ fitting into the following commuting diagram where $p: E \to B$ is a fiber bundle (by the way this is true ...
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### Detail in the proof that sheaf cohomology = singular cohomology

Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of ...
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### Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
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### Trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$

I am trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$ I know that $RP^2$#$T^2$~$RP^2$#$K^2$ and that $X(M$#$N)$=$X(M)+X(N) -2$ where X is the ...
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### Trying to Understand Van Kampen Theorem

Theorem. Let $X$ be the union of two path-connected open sets $A$ and $B$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x_0$ be a point in $A\cap B$ and all fundamental groups will ...
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### Structure space of a commutative ring with unity

I was reading the topic for the discussion of Stone-Cech compactification but stocked at some point: Suppose that $(R,+,.)$ is a commutative ring with unity. Let $\mathcal{M}(R)$ denotes the ...
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### Are all subgroups of the fundamental group of a compact smooth manifold finitely generated?

And if not, is there a way to assign a size to a subgroup by considering the compactness of the corresponding covering space?
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### Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
### degree 1 map $f : M \to S^n$
Let us consider $M$ a closed and connected n-manifold which is also orientable. An exercise in Hatcher claims that for any such $M$ there is a continuous map $f: M \to S^n$ such that it's degree is 1, ...
### Every space $X$ can be identified with the closed subspace of the reduced cone $\operatorname{C}(X)$ of $X$.
For any pointed space $(X,x_0)$, we define the cone $(\operatorname{C}(X),*)$ of $X$ to be the smash product $(X\wedge I, *)$ where the base point of $I$ is assumed to be $0$. The map \$\mathbf{i} : X\...