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

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0
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1answer
34 views

Why are higher homology groups not the abelianizations of higher homotopy groups?

Really the question is exactly the title: Why (conceptually and geometrically if possible) are higher homology groups not the abelianizations of higher homotopy groups?
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2answers
34 views

Covering space of $S^1 \vee S^1$?

Is this a covering space of $S^1 \vee S^1$? I'm not sure what the map from this space onto $S^1 \vee S^1$ does. What is mapped onto which $S^1$?
2
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1answer
26 views

Understanding cellular homology: degree of attaching map of a two cell

I am working towards an understanding of cellular homology as explained here on Wikipedia. To help me I am calculating a simple example: I have two problems: good mathematical notation and actual ...
0
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1answer
35 views

Fundamental Group is free on infinite generators.

This is question 16 of section 1.2 in Hatcher's Algebraic Topology. I have to show that the fundamental group of the space $X$ is free on an infinite number of generators. So here is my approach. ...
2
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1answer
33 views

Explicit form of a lift $\tilde f: \tilde X_1 \to \tilde X_2$ of a continuous map $f: X_1 \to X_2$

This is embarrassingly simple for most, but I am a High School student trying to teach myself, and I am having trouble figuring it out: In the post Basic question about lifting maps to covering ...
2
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1answer
37 views

Homotopy class of maps to a complex projective space

Let $M$ be a closed oriented smooth 4-manifold. Denote by $[M, \mathbb{C}P^{\infty}]$ homotopy classes of continuous maps from $M$ to $\mathbb{C}P^{\infty}$. I would like to know how to show $$ [M, ...
4
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2answers
56 views

Fundamental groupoid

Let $(X,x_0)$ be a pointed topological space. The homotopy groups $\pi_n(X,x_0)=Hom((S^n,s_0),(X,x_0))$ are groups because $S^n$ is a cogroup object in the pointed homotopy category. Removing the ...
2
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1answer
35 views

Universial covering and fundamental group of a space of pairs

Let $M$ be the space of pairs $\{(l,P)|l \subset P \subset R^3\}$ where $l$ is a one-dimensional subspace and $P$ is a two-dimensional subspace of $R^3$. Define a injection $M \rightarrow RP^2 \times ...
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2answers
43 views

What does “modulo a homotopy” mean?

From what I understand, the fundamental group of a topological space $X$ with base point $x_0$ is the set of all equivalence classes of continuous paths in $X$ that start and end at $x_0$. Formally ...
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0answers
37 views

Explicit calculation of homology groups of $S^m\times S^n$ and $\mathbb CP^n$

How can I calculate the homology groups of the complex projective space and the product of two sphere ie $H_k(S^m \times S^n)$ and $H_k(CP^n)$. I have seen some proof using Mayer-Vietoris but since ...
2
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2answers
36 views

Is there Domain invariance for manifolds with boundary in some sense?

It is well known that for manifolds without boundary, there exist a domain invariance theorem in the following form. Theorem. A subspace in an $n$-dimensional manifold without boundary is open if and ...
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1answer
34 views

What does the closure of a subset of a CW-complex look like? Like this?

Introduction Let it be that $X$ is a CW-complex. I practicize the following definition: $X$ is a Hausdorff space and $\mathcal{E}$ is a partition on $X$ such that each $e\in\mathcal{E}$ can be ...
1
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1answer
35 views

Fundamental group Pi1(SU(n)) and Pi2(SU(n))

I need to find the fundamental group $\pi_1(SU(n))$ and $\pi_2(SU(n))$ for all $n$. I don't have any idea.
0
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1answer
22 views

On Hopf invariant

I didn't understand following expression from Hatcher. Let $f: S^{2n-1} \to S^{n}$. If $f$ is a constant map, then $Cf=S^{2n} \lor S^{n}$ and $H(f)=0$ since $Cf$ retracts onto $S^n$.
1
vote
1answer
46 views

Brouwer theorem

Is the Brouwer's fixed point theorem true for the topological space '+' sign(cross)? $$ + = \left( [-1,1] \times \{0\} \right) \cup \left( \{0\} \times [-1,1] \right) $$ I have tried using spencer's ...
1
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2answers
41 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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1answer
19 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 ...
2
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0answers
49 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 ...
-3
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2answers
51 views

Fundamental group of sphere with two holes [on hold]

What is the fundamental group of 2-sphere with two points removed?
2
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0answers
78 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
30 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
20 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
votes
0answers
47 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
38 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
64 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
28 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 ...
3
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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
37 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
votes
1answer
53 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
39 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 ...
0
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2answers
35 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 ...
1
vote
1answer
49 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
49 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
59 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
40 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 ...
1
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0answers
37 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
votes
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
42 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
46 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 ...
0
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0answers
39 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
50 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, ...
6
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1answer
71 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
50 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
54 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
39 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 ...
2
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4answers
53 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
47 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
vote
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 ...