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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4
votes
1answer
30 views

Application of Fixed Point Theorem

Can we prove, if $f:\mathbb{D}^2\rightarrow\mathbb{D}^2$ is a homeomorphism then $f(S^1)=S^1$ and $f(\textrm{int}(\mathbb{D}^2))=\textrm{int}(\mathbb{D}^2)$, using fixed point theorem? I have already ...
0
votes
0answers
20 views

f is null homotopic if and only if (-s,f):cone(C)->D

Actually this question is from Weibel, exercise 1.5.2. Let $f:C\to D$ be a map of complexes. Show that $f$ is null homotopic if and only if $f$ extends to a map $(-s,f):$cone($C$)$\to D$. ...
0
votes
1answer
23 views

Is this a right calculation of Homology groups?

Let $X$ be a circle $S^{1}$ and f a map $f:S^{1} \rightarrow S^{1} $; $f(z)=z^5$. I'd like to calculate homology groups of mapping torus of this space.$$$$ $[X \times I]/\cong$ can be depicted as a ...
7
votes
6answers
158 views

Examples of $\pi_1 (X) = \mathbb{Z}$ [on hold]

I want to know some examples of topological spaces whose fundamental group is isomorphic to set of integers. First, of course i know $\mathbb{S}^1$, and its deformation retract, $\mathbb{R}^2 - \{ ...
4
votes
2answers
31 views

Maps between manifolds with boundary and homeomorphism

Assume we have $f:(M,\partial M)\rightarrow (N,\partial N)$ connected 3-manifolds, not compact, such that $f$ is an homeomorphism onto its image and $f(\partial M)=\partial N$. Can say that $f$ has to ...
2
votes
1answer
24 views

How to tell if a manifold can be embedded as the interior of a compact manifold with boundary?

Some (topological) manifolds can be embedded as the interior of a compact manifold with boundary. Any closed manifold, for example, or any closed manifold with some points removed, and so on. On the ...
0
votes
0answers
32 views

Degree of maps $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$

In the book I am reading right now, it is defined that for a map $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$ the degree is the degree of the direct image cycle $\mu_{*}[\mathbb{P}^1]$. We are ...
0
votes
0answers
34 views

Why “deformation retracts to” instead of “deformationally retracts to”?

First, correct me if I'm wrong: I grammatically understand "X deformation retracts to Y" as "X(subject) deformation(adverbial) retracts to(verb) Y(object)". But I just don't understand why we don't ...
2
votes
2answers
59 views

Is topology invariant under conformal transformation?

Can conformal transformation change the topology of a manifold? In other words, if two manifolds are conformal, should they have the same topology?
1
vote
1answer
39 views

Wedge sum of simply connected spaces is simply connected?

Suppose X and Y are simply-connected. Then their wedge sum is simply connected? My guess is yes, Their wedge sum, $X \vee Y$ is simply connected. But i couldn't get it rigorously. Can you give me ...
2
votes
1answer
21 views

How to prove that $MU$ is an oriented spectrum? A doubt in the proof in Kochman's book

I want to show that the Thom spectrum $MU$ is oriented, namely I want to find a class $x \in \widetilde{MU}^2(\mathbb{C}P^{\infty})$ whose restriction to $\widetilde{MU}^2(S^2)$ is a generator. in his ...
1
vote
0answers
40 views

Homology groups of $S^1 \times (S^1 \vee S^1)$ [duplicate]

How to compute all homology groups of $S^1 \times (S^1 \vee S^1)$? Thank you. I don't see how to define a simplicial structure.
3
votes
2answers
33 views

Non-discrete paracompact space example

Does exist a non-discrete paracompact example where is possible to give a partition of the unity with the functions defined explicitly for a specific not trivial cover of the space?
2
votes
1answer
52 views

Fundamental group of the n-holed torus

I am trying to determine the fundamental group of the n-holed torus I know that the fundamental group of the torus is $\mathbb{Z} \times \mathbb{Z}$ The n-holed torus deformation retracts onto n ...
0
votes
0answers
23 views

Computing Intersection Product in a Cell Complex

Here's the information I have: I have an abstract cell complex that represents some space I am studying, and it is known that the space is orientable. I can identify the sub complexes which ...
1
vote
1answer
32 views

Fundamental group of the complement of $S^1 \cup Z \subset \mathbb{R^3}$

I want to calculate the fundamental group of the complement of this space: This is the complement of $S^1 \cup Z \subset \mathbb{R^3}$ where $S^1$ is the unit circle in the $xy$-plane and $Z$ is ...
4
votes
1answer
62 views

Is there a closed loop in the complex plane such that for any given integer $x$, I can find a point inside the loop that has winding number $x$?

We've been discussing winding numbers in my complex course, and also Alexander polynomials and other invariants on knots in my Alg. Top. course, and the question came to me about the possibility of ...
0
votes
1answer
14 views

How to intuitively see a homeomorphism of the following “triangle” space

Construction of the space: Take two line segments AB and BC and think of them as half open-half closed intervals which are open on the A and C points. Now connect them at B so that you have formed ...
1
vote
1answer
83 views

Homology as categorification of Euler characteristic

I am trying to understand: "Thus, the homology of a manifold M can be seen, in a sense, as a categorification of its Euler characteristic: a more sophisticated and richly structured bearer of ...
3
votes
2answers
158 views

Homotopy Poincaré conjecture - no map inducing the isomorphism on homology

$\newcommand{\Z}{\mathbb{Z}}$ In Terence Tao's notes on page 18, concerning the Poincaré conjecture, he gave the following sketchy proof of the homotopy Poincaré conjecture. Given $M^3$ a 3-manifold ...
3
votes
1answer
57 views

Why does the homotopy lifting property imply that fibers are homotopy equivalent if the base is path connected?

Suppose that $\pi:E \to B$ has the homotopy lifting property, so that for any space Y with a map $f:Y \to E$ and a homotopy $G$ of $g = \pi \circ f$, we have a homotopy $F: Y \times I \to E$ that ...
1
vote
0answers
154 views
+50

$\Delta$-complex structure of the cone and the suspension

I am going around in circles trying to answer the following question: Let $Y$ be a $\Delta$-complex. Describe a $\Delta$-complex structure of its cone $CY=(Y\times[0,1])/(Y\times\{0\})$ its ...
1
vote
1answer
30 views

Relation between path-homotopy classes and path-components

Let us have a topological space $X$. What is the relation between path-homotopy classes and path-components? For example, can we somehow define a map from path-components to path-homotopy classes?
1
vote
1answer
38 views

Building $MU$-spectrum via the definition of a $(B,f)$-structure

I want to construct the them spectrum $MU$ using the definition of spectrum associated to a $(B,f)$-structure. Here are the relevant definitions: A $(B,f)$-structure is a collection of strictly ...
1
vote
1answer
26 views

Induced map on zeroth homology is zero

I am working through some examples on the homology of mapping torii in Hatcher's Algebraic Geometry. One thing that is confusing me is the following: I don't see why the map on the zeroth homology ...
0
votes
1answer
39 views

Understanding the Fundamental Group

Let us have a space $X$. Define the fundamental group $\pi_{1}(X,x_0)$ for some point $x_0 \in X$. If I understood it well, this group contains the path-homotopy classes that are consisting of paths ...
1
vote
1answer
30 views

If $[M]\in H_n(M,\mathbb{Z})$ is a fundamental class for manifold $M$, is $i(M)\in H_n(M,\mathbb{C})$ a fundamental class wrt $\mathbb{C}$?

I wonder if $[M]\in H_n(M,\mathbb{Z})$ is a fundamental class for manifold $M$, is $i(M)\in H_n(M,\mathbb{C})$ a fundamental class wrt $\mathbb{C}$? I.e. does the image of $[M]$ in ...
0
votes
1answer
45 views

Number of non-homotopic diffemorphism form a manifold to itself

What is the name of this invariant, the number of non-homotopic diffemorphism form a manifold to itself. What is this number for the closed ball B^n, and for euclidean space R^n and for the n-sphere?
0
votes
1answer
33 views

Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial hight homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...
1
vote
1answer
20 views

Excercise 1(b) on Zero-dimensional Homology in Munkres

If $\phi:C_0(K)\to \mathbb{Z}$ is an epimorphism such that $\phi\circ \partial_1=0$ then show that $$H_0(K)\cong \frac{ker\phi}{im\partial_1}\oplus\mathbb{Z}.$$ My working is since $C_0(K)$ is ...
0
votes
0answers
31 views

Homology of $X/\{x\sim f(x)\}$ where $f\colon X\to X$

Let $X$ be a space and $f\colon X\to X$ a continuous map. What tools do we have to compute the homology $H_n(X/\sim)$ where $\sim$ is defined by $x\sim f(x)$? Relative homology was my first though, ...
2
votes
2answers
36 views

Intuition of product spaces

So I have a product space of the form: $X=X_1 \times \ldots \times X_n$ and I take two elements of it, say $x=\{x_1,\ldots,x_n\}$ and $x'=\{x_1',\ldots,x_n'\}$. Now suppose I take the following ...
0
votes
0answers
23 views

Does the induced actions of a continuous map on the covering space determine it?

If I have continuous maps $f,g : X \rightarrow Y$ between topological spaces which induce the same actions of the fundamental group of X on the covering space of Y, are they necessarily homotopic?
4
votes
2answers
66 views

Klein bottle in $\mathbb{R}^4$ does not have a couple of normal vector fields

For an orientable 2-manifold in $\mathbb{R}^4$, it's somewhat obvious that if the manifold has a normal vector field then it has a pair of normal vector fields. I am trying to understand why it is ...
0
votes
0answers
31 views

Reason why “Cut and Paste” of Fundamental Polygon is allowed

Background: I am studying this question by Hatcher: What familiar space is the quotient $\Delta$-complex of a 2 simplex $[v_0,v_1,v_2]$ obtained by identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$, ...
1
vote
1answer
30 views

Isomorphism on top cohomology implies isomorphism on homology

Let $F$ be a finite field (for example I could take $\mathbb{Z}_2$) and $f:X\longrightarrow Y$ a continuous map between compact, orientable and connected manifolds of dimension $n$. Suppose I have an ...
0
votes
1answer
29 views

Constructing a map $H^{k}(M,\mathbb{Z})\to H^{k}(M,\mathbb{C})$

I read that on a compact oriented manifold, there is a map $$H^{k}(M,\mathbb{Z})\to H^{k}(M,\mathbb{C}).$$ I want to be sure that I have the right map in mind. We don't have an inclusion, since ...
1
vote
0answers
25 views

Homology and triangulation of open surfaces

For example I have an open disk, or an open annulus. How do I triangulate open surfaces to find their (simplicial) Homology? Well, I know that open disk and closed disk are both homotopic to a ...
2
votes
1answer
28 views

Some basic question on pasting map from a square to a Klein bottle and homology

Consider a square $S$ which edges identified as follows Let $K$ be a Klein bottle and $p:S\to K$ be pasting map. Let $X$ be the image of the interior of $S$ under $p$ and let $Y$ be the image of a ...
2
votes
1answer
32 views

Bott&Tu Definition: "Types of Forms:

In Bott&Tu's well-known book "Differential forms in Algebraic topology", they note -(p34): every form on $\mathbb{R}^n \times \mathbb{R}$ can be decomposed uniquely as a linear combination of two ...
0
votes
0answers
33 views

Are complex subvarieties cycles in the sense of singular homology?

Given a $p$-codimensional complex subvariety $Z\subset M$ of a non singular complex projective variety $M$ of dimension $n$ we can define an element $$\int_\hat{Z}i^*\in ...
1
vote
0answers
31 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
7
votes
1answer
90 views

$S^3\times \Bbb CP^\infty$ is not homotopy equivalent to $\left(S^1\times \Bbb CP^\infty\right)\big/\left(S^1\times \{x_0\}\right)$

Both $S^3\times \Bbb CP^\infty$ and $\left(S^1\times \Bbb CP^\infty\right)\big/\left(S^1\times \{x_0\}\right)$ have cohomology ring isomorphic to $\Bbb Z[a]\otimes \Lambda[b]$ with $|a|=2$ and ...
6
votes
1answer
90 views

Non-injective monomorphisms

I am reading Borceux, vol. 1, I found this example at page 27: we consider the category whose object are the pairs $\langle X,x\rangle$ where $X$ is a topological space and $x$ a point of $X$ (base ...
1
vote
0answers
34 views

Relative Homology (Question about Example 2 in Munkres)

I have no problems for $p=0$ case and for $p\geq 2$ it is quite obvious since $C_p(K,v)=C_p(K),\forall p\geq 2$. Now the tricky is for $p=1$. Since the elements of the kernel now not anly map to $0$ ...
1
vote
1answer
24 views

Understanding proof of Universal coefficient theorem for cohomology

I am working through Cohomology chapter on Hatcher's book and I am having trouble with the proof of Universal Coefficient theorem for Cohomology. To be concrete I don't understand the last part of the ...
0
votes
2answers
45 views

Isomorphism in fundamental group implies isomorphism is homology

Let $X$ be a connected space and $f:X\longrightarrow X$ a map. Suppose $\pi_1(X)$ is an abelian group and that $\pi_1(g):\pi_1(X)\longrightarrow\pi_1(X)$ is an isomorphism. I know we can deduce that ...
0
votes
0answers
11 views

Continuous maps to $S^n$ without antipodal pairs are homotopic [duplicate]

Let $S^n$ denote the unit sphere in the Euclidean space $\Bbb R^{n+1}$, $X$ a topological space, $f,g:X\to S^n$ are both continuous and there doesn't exist $x\in X$ such that $f(x)=-g(x)$, show ...
0
votes
0answers
24 views

Homotopy sets for a pushout of spaces (Seifert-van-Kampen?)

my problem is the following: I have two bordisms $M : \Sigma_0 \to \Sigma_1$ and $M' : \Sigma_1 \to \Sigma_2$, so I can glue them along $\Sigma_1$ to get $M'\circ M$. The manifold $M'\circ M$ is the ...
3
votes
1answer
35 views

Does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form?

Question: On a $C^\infty$ manifold, does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form? Motivation: This result holds for $C^1$ closed 1-forms on a ...