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

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3
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1answer
70 views

Finding the degrees of the attaching map of the $2$-cell of the torus

I am trying to calculate the degrees of the attaching map of the two cell of the torus. I have the following cell structure: The $2$-cell is $e_2$ and the $0$-cell (all four corners) is $e_0$. I ...
2
votes
2answers
154 views

Book for Algebraic Topology- Spanier vs Tom Dieck

A number of times, questions have been asked on this website about good books on Algebraic Topology and the responses have been very valuable. However I need some more specific advice in this matter. ...
2
votes
1answer
65 views

Is this a covering space of $S^1 \vee S^1$?

Is the following a covering space of $S^1 \vee S^1$ ? It would appear so since there is no point that has more than 2 incoming or outgoing arrows. It seems that the potential covering map $p:Y\to ...
2
votes
0answers
22 views

CW approximation of $n$-connected space

I want to prove the following lemma: Let $X$ be a n-connected space. Then there exists a CW-approximation $f:K\rightarrow X$ such that $K$ has trivial n-skeleton. What I have done so far: By ...
3
votes
0answers
94 views

Fundamental group of connected sum for non-orientable manifolds

For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$. As far as I understand, for non-orientable ...
3
votes
1answer
52 views

Obtaining the Möbius strip as a quotient of $S^1\times[-1,1]$

I am trying to obtain the Möbius strip as a quotient of $S^1\times[-1,1]$, where $S^1$ is, of course, the circle. My definition of Möbius strip is the quotient of the square $[0,1]\times[0,1]$ by the ...
1
vote
1answer
40 views

Surjection of the fundamental group of a manifold onto a free group induces a map onto a wedge of circles, why?

Why does a surjective map $\pi_1(M)\twoheadrightarrow F$ of the fundamental group of a manifold $M$ onto a free group $F$ over $n$ generators induce a continuous map $M\twoheadrightarrow\bigvee^n S^1$ ...
1
vote
1answer
25 views

The Projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})$

So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a ...
1
vote
2answers
59 views

$f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$

Let $f:S^{2n}\rightarrow S^{2n}$ continuous. Then there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$. I am having a hard time finding a starting Point. Thank you
1
vote
0answers
38 views

Spin structures, frame bundles, and trivializations over the 2-skeleton

While reading an introduction to Spin- and Spin$^{\operatorname{c}}$ structures (found here), I encountered the following definition: Let $E\to X$ be an oriented $\mathbb{R}^n$-bundle over a CW ...
0
votes
1answer
57 views

Properly discontinuous action: equivalent definitions

Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = ...
1
vote
3answers
118 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?
1
vote
2answers
73 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 $S^1$?
3
votes
1answer
46 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
votes
1answer
45 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
votes
1answer
50 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
votes
1answer
45 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
votes
2answers
76 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
votes
1answer
71 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 ...
0
votes
2answers
44 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 ...
2
votes
2answers
58 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 ...
1
vote
1answer
47 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
vote
1answer
58 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
votes
1answer
33 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
56 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
vote
2answers
49 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
votes
1answer
23 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
votes
0answers
61 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 ...
2
votes
1answer
35 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
votes
0answers
28 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
51 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
votes
1answer
48 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 ...
5
votes
1answer
75 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 ...
0
votes
0answers
36 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
votes
1answer
47 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 ...
1
vote
2answers
43 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 ...
3
votes
1answer
69 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
votes
1answer
48 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
votes
2answers
39 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
52 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 ...
1
vote
3answers
59 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
votes
0answers
66 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
votes
1answer
31 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 ...
4
votes
0answers
46 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
42 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
68 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
votes
1answer
50 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
votes
1answer
49 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 ...
2
votes
1answer
53 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, ...
5
votes
1answer
170 views

The cone of a topological space is contractible and simply connected

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 ...