Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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1answer
21 views

What's a Labeling Scheme?

I have to learn how to solve problems like the following in the next two weeks: Let $X$ be the quotient space obtained from an 8-sided polygonal region P by pasting its edges together according to ...
0
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0answers
8 views

Does Alexander Duality commute with inclusion?

This is a follow up to this question I asked previously: Alexander Duality for Relative Homology I am working with two compact pairs of spaces $(A,B)$ and $(A',B')$, where $A'\subset A$ and ...
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0answers
31 views

Unique Path lifting of covering map

Let $p:E\rightarrow B$ be a covering map (in particular $p$ is a fiber bundle with discrete fiber). We want to prove the following: Given a commuting diagram of the following form: $\{0\}\rightarrow ...
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2answers
41 views

Fundamental group of sphere with two holes [on hold]

What is the fundamental group of 2-sphere with two points removed?
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0answers
40 views

Reference for proof of even cohomology group of sphere $S^{2n}$ is trivial

I am new to the idea of cohomology. I want to understand the proof for The even cohomology groups of the sphere $H_{deR} ^{2k} (S^{2n})$ for $ 0 < k < n$ are trivial. Any suggestions for ...
2
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1answer
28 views

Find homotopy fibers of $\mathbb{R}P^1 \hookrightarrow \mathbb{R}P^\infty$ and $\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^\infty$

I found a following tasks in my algebraic topology notes: Find homotopy fibers of $\mathbb{R}P^1 \hookrightarrow \mathbb{R}P^\infty$ and $\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^\infty$. For a ...
2
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0answers
19 views

The images of two non homotopic to identity maps intersect

How could one prove that images two maps $f,g:\mathbb RP^4 \to \mathbb RP^7$ which are not homotopic to trivial map have nonempty intersection.
3
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0answers
45 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
4
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1answer
35 views

Show that $M^n-N^{n-1}$ has exactly two components with $N^{n-1}$ as the topological boundary of each.

This is the problem 6.8.1. from "Topology and Geometry" by Glen E. Bredon. The problem is, If $M^n$ is a connected, orientable, and compact $n$-manifold with $H_1(M^n;\mathbb{Z}) = 0$ and if ...
4
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1answer
61 views

Degree of a map $S^0 \to S^0$

By definition the degree of a map $f: S^n \to S^n$ is $\alpha \in \mathbb Z$ such that $f_\ast(z) = \alpha z$ for $f_{\ast}:H_n(S^n) \to H_n(S^n)$. What is the definition of the degree of $f: S^0 \to ...
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0answers
25 views

Cellular homology $d_n$ definition by example

I'm trying to understand cellular homology. Consider the following diagram taken from these lecture notes: I'm trying to understand what the maps $d_n$ are. As I understand $H_n(X^n, X^{n-1})$ is ...
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1answer
45 views

What to do when this theorem can't be applied: How to calculate $H_1$?

Consider the following theorem (Lee's book on topological manifolds, page 369): (Homology Effect of Attaching a Cell) Let $X$ be any topological space and let $Y$ be obtained from $X$ by attaching a ...
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2answers
36 views

2-dimesional cell complexes with fundamental group isomorphic to the following.

I have been asked to give examples of 2-dimensional cell complexes whose fundamental group isomorphic to the following $$ \Bbb Z_4 * \Bbb Z_5$$and $$\Bbb Z_4\times \Bbb Z5$$ I know in the first ...
2
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1answer
43 views

How do modern categories of spectra manage to avoid “cells now, maps later”?

In Adams' definition, a map $f: X \to Y$ of CW spectra consists of a cofinal subcomplex $X'\subseteq X$ and maps $f_n: X'_n \to Y_n$ that commute with the structure maps. This definition is reproduced ...
2
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1answer
36 views

Covering map is proper $\iff$ it is finite-sheeted

Prove that a Covering map is proper if and only if it is finite-sheeted. First suppose the covering map $q:E\to X$ is proper, i.e. the preimage of any compact subset of $X$ is again compact. Let ...
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2answers
34 views

What does it mean when people say the co fiber $C_f$ of $f: X\rightarrow Y$ does not dependent on f functorially in homotopy category?

Want to form the mapping cone of a map $f: X\rightarrow Y$ in the homotopy category. I am hoping that some one can give easy examples to show that mapping cone $Y \cup_f CX$ does not dependent on f ...
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1answer
46 views

Degree of this attaching map — or how to define this attaching map?

Consider the cell complex consisting of two zero cells $e_0^1, e_0^2$ connected by two 1 cells $e_1^1,e_1^2$ with one 2 cell $e_2$ in the middle (Picture: Imagine $S^1$ with one $0$-cell at the north ...
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3answers
48 views

Difference between cellular and simplicial homology

Can someone tell me if there is any difference between cellular and simplicial homology? It seems to me that when I calculate for example the homology groups of the torus it makes no conceptual ...
2
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0answers
52 views

Dual of path in a space.

Is there a notion dual to the notion of a path in a topological space? Given that a path in a space $X$ is a continuous function from the interval $[0, 1]$ to X, what would the dual of this notion be, ...
0
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1answer
30 views

Steps involved when showing this induced map on homology is welldefined

I am showing that $H_0(X,R)=R$ when $X$ is a path-connected topological space. Let the zero boundary map be $\partial_0 : C_0(X) \to R$, $c \mapsto 0$. Define a map $\varphi : C_0(X) \to R$ by ...
3
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0answers
39 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
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0answers
34 views

$E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering

Let $q:E\to X$ be a covering map. Then $E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering. My question is regarding the $"\implies"$ direction: If $E$ is compact, then ...
4
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2answers
47 views

Two-sheeted covering of the Klein bottle by the torus

Prove that there is a two-sheeted covering of the Klein bottle by the torus. OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows: Then there are ...
2
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1answer
40 views

If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
0
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1answer
45 views

A question regarding singular homology (The geometrical interpretation of cycles in singular homology)

From Hatcher's Algebraic topology : Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite Δ-complexes. To ...
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0answers
38 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
2
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1answer
49 views

Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, ...
2
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0answers
68 views

A more detailed, rigorous proof that a suspension space is not necessarily contractible

Is my answer/proof correct? Please help me make my proof more rigorous and detailed. I need everything to be absolutely clear. Question: Let $X$ be a topological space. The suspension of $X$, ...
6
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1answer
70 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
2
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1answer
49 views

Homology of universal cover of $S^1 \vee S^1 \vee S^2$ is not the same as homology of $\Bbb R^2$

I want to show, that although the homology groups of $X :=S^1 \vee S^1 \vee S^2$ and the torus $T^2$ are isomorphic, the homology groups of their universal covers are not. Let $U_X$ be the universal ...
3
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2answers
53 views

Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix. Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
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2answers
35 views

How do I prove this using van-Kampen theorem informally ? (2)

First of all, I feel really sorry to keep asking these questions without a try, but it is because I can't try.. Really if someone wants to see my lecture note I can show it.. Statement of the theorem ...
4
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3answers
34 views

How do I informally prove this using Van Kampen theorem?

Let $X$ be the space obtained from two tori $S^1\times S^1$ by identifying a circle $S^1\times\{x_0\}$ in one torus with the corresponding circle $S^1\times\{x_0\}$ in the other. Calculate ...
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4answers
52 views

How do I prove that a finite covering space of a compact space is compact?

Let $C$ be a finite sheeted covering space of compact space $X$. How do I prove that $C$ is compact? Someone please give me a proof sketch.. Let $p:C\rightarrow X$ be a covering map. Let ...
6
votes
2answers
43 views

Example of closed orientable manifold with a nonzero vector field, but not parallelizable

Can you give a simple example of a closed orientable manifold with an everywhere nonzero section of its tangent bundle, but where the tangent bundle is not trivial? If the manifold is not orientable, ...
1
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1answer
25 views

How do I prove this winding number is not zero?

Let $\alpha:[0,1]\rightarrow S^1:t\mapsto e^{2\pi it}$ be a path. Let $f:S^1\rightarrow S^1$ be a continuous map such that $-f(x)=f(-x)$ on $S^1$. How do I show that the winding number $Wnd(f\circ ...
2
votes
1answer
32 views

Non-connectedness in the plane

Let us have open $V \subset \mathbb{R}^2$ and $x\in V$. Now, how can I prove that the quotient space $V\backslash\{x\}$ is not simply connected? Pictorially I understand it as the failure of a loop to ...
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0answers
21 views

Is this map homotopically trivial?

There is a map of cosimplicial spaces $f^*: X^*\to Y^*$ such that for every $n$, the map $f^n:X^n\to Y^n$ is homotopically trivial. Is this true that the induced map $a(f):a(X^*)\to a(Y^*)$ is also ...
2
votes
1answer
35 views

Is every open set in a base space evenly covered?

Let $C$ be a covering space of $B$. Then, does every open set in $B$ evenly covered by a covering map? This must be false but I cannot find a counterexample.. Please help
2
votes
1answer
69 views

Prove that these loops are homotopic [duplicate]

Let $G$ be a topological group with identity element $e$. Let $f,g: (S^1, (1,0)) \to (G,e)$ be loops in $G$ with base point $e$. We define $f * g: (S^1, (1,0)) \to (G,e)$ by $$f * g(s) = f(t) \cdot ...
2
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1answer
28 views

Unit close disc to prove a matrix algebra identity?

I need to prove that every $3 \times 3$ matrix with real positive entries has one eigenvector with a positive eigenvalue. Now, how do I prove this using the fact that the set $B=\{x=(x_1,x_2,x_3)\in ...
2
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1answer
26 views

Alexander Duality for Relative Homology

Is there a formulation for Alexander Duality for pairs of spaces $(A, B)$ such that $A\subset B\subset S^n$? I can't find a reference for this anywhere, but I think it is as follows, which I arrived ...
2
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1answer
50 views

Kernel of induced map between singular chain groups

Let $p : \widetilde X \to X$ be a two-sheeted cover. This induces $p_\sharp : C_n(\widetilde X; \mathbb Z_2) \to C_n(X; \mathbb Z_2)$. I can show that $p_\sharp$ is surjective by noting that every ...
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1answer
44 views

Fundamental group of composition of function

Let Y be a simply connected space, and let f : X → Y and g : Y → Z be continuous functions. What is π1(g ◦ f )? Prove your answer. I think the fundamental group of the composition would be the ...
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2answers
74 views

Fundamental group is a homotopy invariant

I am a newbie to topology and am not able to understand how to attack this problem: Any hints would be appreciated Assuming that: $$ f \sim g \Rightarrow \pi_1(f) = \pi_1(g). $$ Prove that the ...
2
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2answers
61 views

Map from $n$-sphere to $n$ dimensional torus

Let $n\ge 2$. How can you prove that for every continuous $f:S^n\to T^n$, the induced map on singular homology $f_\star:H_n(S^n)\to H_n(T^n)$ is the zero map? Here, $S^n$ is the $n$ dimensional ...
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0answers
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+150

Action of $\operatorname{Aut}(G)$ on the Borel construction

I am interested into the (say real) regular representation $\rho$ of $G=(\mathbb{Z}/p)^n$. Considering the universal vector bundle $EG\rightarrow BG$, the Borel construction with the regular ...
2
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1answer
51 views

Union of simply connected spaces at a point not simply connected

I came across this example Spanier's Algebraic Topology book (by way of the Munkres Topology book). I kind of have an intuitive idea of why the space isn't simply connected but can't figure out a ...
2
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0answers
52 views

covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
2
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0answers
37 views

Van Kampen Theorem for a Certain Square

Take a square with all the edges identifies. Choose a point $x$ on the boundary of this square. Take a small neighborhood $B_\epsilon(x)$ of this point. I want to compute $\pi_1(B_\epsilon(x) ...