# 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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### Fundamantal group of a regular covering space

Let $B$ be the space of figure $\infty$ (with $x$(the red circle) and $y$(the black one) as generators) and $E$ its covering space (in the picture below). let $P_{*}: \Pi(E,a) \to \Pi(B,b)$ be the ...
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### is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
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### Two deformation retractions (onto $A$) are homotopic (rel $A$).

This is a question from Hatcher's Algebraic Topology (Chapter 0, Question 13): 13. Show that any two deformation retractions $r^0_t$ and $r^1_t$ of a space $X$ onto a subspace $A$ can be joined by ...
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### Homotopy between two functions to a circle.

Suppose $f,g: X\to S^1$ are such that $f(x)\neq -g(x)$ for any $x\in X$. I need to construct a homotopy between these two functions. Now, the fact that $f(x)\neq -g(x)$ guarantees that there is always ...
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### Non-homotopy equivalent spaces with isomorphic fundamental groups

I know that if two spaces $X,Y$ have the same fundamental group : $\pi_1(X,a) \cong \pi_1(Y,b)$ does not imply that they are homotopy equivalent. But I can't find an example. I was thinking about ...
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### Local dimension of graph embedding

I am trying to find a way to characterize the dimension of the smallest space into which a (neighbourhood of) a graph $\Gamma = (V, E)$ may be embedded. Although in the end my goal is to identify what ...
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### Intuition on Whitney–Graustein theorem

According to the Whitney–Graustein theorem, two regular curves are regularly homotopic if and only if their winding numbers are the same. Suppose I have a circular curve but with an extra loop so ...
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### How do Chern classes behave under connected sums?

I often see people talking about Chern classes of manifolds like $\mathbb{CP}^2 \# \mathbb{CP}^2$, or $\mathbb{CP}^2\#\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$, where $\#$ denotes connected sum and the ...
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### Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
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### What is a 3D winding number?

This paper mentions the term "3D winding number". Its abstract says: "We develop a new formulation, mathematically elegant, to detect critical points of 3D scalar images. It is based on a topological ...
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### Inverse question: Recognizing when a phenomenon behaves like an algebra

I might be phrasing this question incorrectly, but my students asked me about it and I did not have a good answer. I am in statistics and I look at all sorts of social data, network data, climate data,...
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### Compactness and Poincare duality

I am reading Appendix B in Fulton's Young Tableaux about Borel-Moore homology. In particular, I'd like to understand why for compact manifolds the Borel-Moore homology groups are isomorphic to ...
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### a Problem about Degree of Map between Spheres

When I read the book "Algebra Topology-A First Course", I find a problem. It is on the Page 97, Exercise (16.15). Problem We define $f$, $g\colon \mathbb{S}^{n}\to\mathbb{S}^{n}$ to be orthogonal ...
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### Are there degree-1 maps from $S^2 \times S^3 \rightarrow S^5$ or from $S^5 \rightarrow S^2\times S^3$?

This is a question from a past qualifying exam I am stuck on: For a smooth map $f:M\rightarrow N$ between smooth, compact, oriented $n$-manifolds, the degree of $f$ is the unique integer $k$ such ...
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### $GL(n,\mathbb{C})$ is semi-locally simply connected.

How can we show that $GL(n,\mathbb{C})$ is semi-locally simply connected.
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### existence of double covering [duplicate]

Let $M$ be a manifold , and $\pi_1(M)=\mathbb{Z}$. then can we say, the double covering of $M$ exists and is unique?
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### Loops in $RP^2$

We know that $\pi_1(RP^2)=Z_2.$ How do non-trivial loops in $RP^2$ look like? (If $RP^2$ is the upper hemisphere $D$ with antipodal points of $\partial D$ identified)
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### Path or One-Dimensional Manifold

The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional ...
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### Self-contained text on characteristic classes

I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and ...
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### Orbit space of S3/S1 is S2

I'm having trouble finishing this homework assignment. I did the first part by showing that the orbits are invariant: every element from the same $(S^1(z_1, z_2) \in S^3/S^1)$ is mapped to the same ...
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### Deleted join of simplicial complex

Give an example of simplicial complex $k$ such that the deleted join of $k$ is homeomorphic with $S^1\times [0,1]$, i.e: $$k^{*^2}_{\bigtriangleup}\cong S^1\times [0,1]$$
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### Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
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### Why $\operatorname{im} \delta_{-1}$ contains two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic function?

In the paper, on page 21, line 15-20. It is said that $B^0=\operatorname{im} \delta_{-1}$ is the one dimensional space containing two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic ...
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### Are space of paths between two different points and space of pointed loops only homotopy equivalent? What about smooth case?

Let $X$ be a path-connected CW-complex and $x$, $y$ points in $X$. Any choice of a path between $x$ and $y$ provides maps (in both directions) between the space $L(x, y)$ of paths from $x$ to $y$ and ...
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### Properties of $\pi_n$ from a category theoretical point of view

This will be a more open question. I would like to have a better understanding about how the homotopy group functors behave with different constructions such as limits or colimitis. Some examples: ...
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### coproduct of base and fiber is weakly equivalent to total space of a fibration in stable model category

Let $C$ be a proper pointed model category such that for any $X, Y \in C$ the natural morphisms $$QX \coprod QY \to X \coprod Y \to RX \times RY$$ are all weak equivalences (here $Q$ and $R$ are ...
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### mayer vietoris homework

Let $X=A\cup B\ \ A,B$ are open and $A\cap B$ is contractible. Prove that $H_i(A\cup B)\equiv H_i(A)\oplus H_i(B)$ for $i\geq 2$. I think about using Mayer Vietoris sequence but I don't know how to ...
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### A (somewhat) conceptual proof that the boundary of a fundamental class of a manifold with boundary goes to a fundamental class?

In this set-up, let M be a compact n-dimensional manifold with boundary $\partial M \neq \emptyset$. Assume that M is orientable, and that $[M] \in H_n(M,\partial M;R)$ is the fundamental class of M....
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### Differential closed form

Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form: $$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$ in some ...
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### Fundamental group of CW-Complex only depends on 2-Skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1 (X)$ only depends on the $2$-Skeleton of $X$ for any $CW$-Complex $X$. I am very ...
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### What are simplicial topological spaces intuitively?

(NOTE: I reposted the question to MO. Please answer there.) I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial sets are ...
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### The singular homology and cohomology of a topological space with coefficients in a zero characteristic field.

I have a field with zero characteristic, like $K=\mathbb{C},\mathbb{R}$ and I want to show that the homology groups and cohomology groups with coefficients in these fields satisfy: H_n(X,K) \approx ...
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### Definition of star in a simplicial complex

Given a simplicial complex K and a collection of simplices S in K, the star of S is defined as the set of all simplices that have a face in S. Now consider the following picture (from wikipedia): ...