# 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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### Fixed-point free map of the 2-sphere which has order 4

The antipodal involution of $\mathbb{S}^2$ clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me?
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### Steenrod Operations an algebraic Approach

Assume that $q=p^{r}$, where $p$ is a prime either 2 or odd and $\mathbb{F}_{q}$ is a Galois field and $V$ a finite dimensional $\mathbb{F}_{q}$-vector space. Then due to Larry Smith in this http://...
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### How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$?

How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$? In general is well known that $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2, ~ n \ge 2.$ But how to show this assertion? I have a few knowledge about ...
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### Two sheeted covering projection

In Hatcher on page 144, example 2.42, I see $RP^n$ described as a CW structure with one cell $e^k$ in each dimension $k\leq n$, and the attaching map for $e^k$ is the 2-sheeted covering ...
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### Cohomology Group basis

I'm reading a text on Complex Torus and Abelian Variety and at a time is written as follows: The cohomology group $H^{1}(T,\mathcal O_{T})$ has a basis $w_{j}=d\overline{z}_{j}, j=1,2,...,g,$ as ...
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### Restriction of a $C^{\infty}$ vector bundle over a regular submanifold.

This question is about the content of page 134 of Tu's An Introduction to Manifolds. A $C^{\infty}$ vector bundle or rank r is a triple $(E,M,\pi)$ consisting of manifolds $E$ and $M$ and a ...
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### Universal Abelian Covering Space of genus two surface [on hold]

Let M be a surface M, i am concerned with Abelian covers. These are the covering spaces for which the deck group is Abelian. The largest such cover corresponds to the commutator subgroup of the ...
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### Let $X$ be the union of a torus with an interval that meets the torus as shown. Use Van Kampen to find a presentation for the group.

I need to come up with some kind of cell structure here right? How can I do this?
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### Classifying space of $GL_{n}(\mathbb{F})$?

I was looking for the classifying space of the general linear group $GL_{n}(\mathbb{F})$ over a field (of characteristic either zero or positive, finite or infinite), but unfortunately I didn't manage ...
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### Topology of complex projective plane

It is well known there are two ways to construct topology of $\mathbb{C}P^n$: quotient space of $S^{2n+1}$ by identifying $x$ with $\lambda x$, where $\lambda$ is complex nonzero constant. According ...
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### Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
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### Intuition behind CW complexes

These things are the bane of my existence in mathematics. I feel that I can't find any clear examples of these things anywhere. This is a vague question, but how exactly do we intuitively visualize ...
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### Fundamental group of hole-punched torus with boundary identification

I'm trying to find the fundamental group of $Y$ obtained from the torus by removing a small disk and identifying the boundary with the torus meridian. Here's my idea. The torus has the polygon ...
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### Fundamental group of $S^{1}$ unioned with its two diameters

Is my solution correct? Call $X$ the riflescope space (I made this name up). I let $p$ be the point of intersection of the two diameters, and $q$ be the right point of intersection of the horizontal ...
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### Obstruction theory vs. homotopy lifting property of Serre fibration

The obstruction to obtaining a lifting to the total space $E$ of a Serre fibration $E \to B$ of a map $X \to B$ can be derived by assuming a CW complex structure for $X$ and examining the obstruction ...
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### Group bundles for topological spaces without universal cover

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. In this context I have two related questions: There are two approaches in defining Homology with local coefficients ...
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### Bott and Tu compact cohomology of the circle “differential forms in Algebraic Topology”

On page 27 of that book, it is claimed that the inclusion map $\delta$ which maps a form from the non-empty intersection of two open covers of the circle to the disjoint union of those covers has a ...
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### Use van Kampens theorem to compute the fundamental group of a torus with a ball attached via a map

Use van Kampens theorem to compute the fundamental group of the following space: $A$ is a torus with an open disk $D$ removed. Let $f:\partial B \rightarrow \partial A$ be a map from the boundary of a ...
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### Local Properties of Immersions and Submersions

This is from Bredon's Topology and Geometry, pp 83~83. He's definitions of immersion and submersion are the following: Now, in 7.3 Theorem, he explains that (using the implicit function theorem) if ...
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### branched cover of projective line over rationals

I was reading something when I came across the following phrase "branched cover of projective line over rational " . To understand what does author mean , I started reading , Now I know about ...
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### Fundamental group of quasi circle(page 79 ,Hatcher) is trivial

I was trying to show that fundamental group of quasi circle(page 79 ,Hatcher) is trivial.I can understand that every loop is precisely zero loop because for any loop if it start at some point it has ...
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### Broken line is NOT diffeomorphic to the real line

This is from Bredon's Topology and Geometry, page 71. This comes right after the very definition of differentiable manifold, so I think no use of tangent space or 'differential' is permitted. (Bredon ...
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### Surjective map between fundamental group of surfaces

Let $f\colon S_m\rightarrow S_n$ be a continuous map of degree $\pm1$. Then the induced morphism $f_\bullet \colon \pi_1(S_m) \rightarrow \pi_1(S_n)$ is onto. How can I prove this? I know that the ...
### Second homotopy of $S^1\vee S^2 \vee T^2$
How can I prove that the second homotopy group of $S^1\vee S^2 \vee T^2$ is infinitely generated? I know that the second homotopu groups of $S^2$ and $T^2$ are finitely generated, so is kind of ...