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

learn more… | top users | synonyms (1)

3
votes
1answer
43 views

Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
1
vote
1answer
20 views

Trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$

I am trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$ I know that $RP^2$#$T^2$~$RP^2$#$K^2$ and that $X(M$#$N)$=$X(M)+X(N) -2$ where X is the ...
1
vote
1answer
40 views

Proving equality of homology of a product

I'd like to prove the following equality: $$H_i(X \times S^{n}) = H_i(X) \times H_{i-n}(X) $$ For $n=0$ it's pretty obvious, hence I'll use induction.Dividing $S^{n}$ into two hemipsheres ...
0
votes
1answer
29 views

Induced homomorphism from homology group of circle to homology group of $\mathbb{R^2-}0$ is trivial

Let $C_r$ be a circle of radius $r$ in complex plane, and let $f:C_r\to\mathbb{R^2}-0$ defined by $f(z)=z^n+a_{n-1}z^{n-1}+...+a_0$ and suppose that it has no zero on and inside the circle $C_r$. ...
1
vote
2answers
61 views

A Natural Question When Reading Van Kampen Theorem

Let $A$ and $B$ be path connected open subspaces of a topological space $X$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x$ and $y$ be two points in $A\cap B$. Let $\gamma$ and ...
2
votes
1answer
48 views

Trying to Understand Van Kampen Theorem

Theorem. Let $X$ be the union of two path-connected open sets $A$ and $B$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x_0$ be a point in $A\cap B$ and all fundamental groups ...
0
votes
0answers
24 views

Structure space of a commutative ring with unit

I was reading the topic for the discussion of Stone-Cech compactification but stocked at some point: Suppose that $(R,+,.)$ is a commutative ring with unity. Let $\mathcal{M}(R)$ denotes the ...
1
vote
0answers
48 views

Are all subgroups of the fundamental group of a compact smooth manifold finitely generated?

And if not, is there a way to assign a size to a subgroup by considering the compactness of the corresponding covering space?
1
vote
0answers
44 views

Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
0
votes
3answers
48 views

degree 1 map $f : M \to S^n$

Let us consider $M$ a closed and connected n-manifold which is also orientable. An exercise in Hatcher claims that for any such $M$ there is a continuous map $f: M \to S^n$ such that it's degree is 1, ...
1
vote
2answers
41 views

Every space $X$ can be identified with the closed subspace of the reduced cone $\operatorname{C}(X)$ of $X$.

For any pointed space $(X,x_0)$, we define the cone $(\operatorname{C}(X),*)$ of $X$ to be the smash product $(X\wedge I, *)$ where the base point of $I$ is assumed to be $0$. The map $\mathbf{i} : ...
1
vote
1answer
24 views

Showing that for $S^1 \subset \mathbb C$, the induced homomorphism of $f_n = z^n$ corresponds to multiplying by n

I'm just starting to learn algebraic topology and was doing some simple exercises. Let $S^1 \subset \mathbb C $ be the circle as a subset of the complex numbers (with $\pi(S^1, 1_\mathbb C) \approx ...
0
votes
1answer
21 views

Euler characteristic of the projective plane (using embedding diagram)

Make the square into the projective plane $\mathbb{P}$ by identifying edges and compute the Euler characteristic by embedding the following graph onto the surface: Here is my diagram of the ...
0
votes
1answer
26 views

Triangulation of double-holed torus to calculate fundamental group

Show that the fundamental group of the double-holed torus is given by: $\pi_1=<a, b, c, d | aba^{-1}b^{-1}=cdc^{-1}d^{-1}>$ I have ended up with the following identification diagram ...
1
vote
1answer
47 views

The fundamental group of a point is $1$

Show that the fundamental group of the point space $p$ is given as $\pi(p, w_0)=1$ where $w_0$ is the base point This is probably somewhat trivial, but I am looking for a proof. I am familiar ...
0
votes
2answers
40 views

Is path-connected a homotopy property of toplogical spaces?

$X$ and $Y$ are homotopy equivalent so there are maps $\alpha: X \rightarrow Y$ and $\beta : Y \rightarrow X$ whose composites satisfy : $\beta\alpha \simeq id_X$ and $\alpha\beta \simeq id_Y$ $X$ is ...
-1
votes
1answer
41 views

Defining a homotopy between maps transforming coffee mug to a donut

Assume that we have two topological spaces $X$ and $\mathbb{R}^3$ and two continuous maps $f_1,f_2$ such that $f_1 :X \rightarrow \mathbb{R}^3$ and $f_2 : X \rightarrow \mathbb{R}^3$. My first ...
1
vote
3answers
28 views

Prove that Euclidean distance and Uniform norm generate the same topology

I'm teaching myself topology using a book I found. I need help proving the following. Suppose $M$ is the unit disk in $\mathbb{R}^2$, $d$ is the Euclidean distance and $d'$ is the Uniform norm. ...
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
21 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
163 views

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

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
32 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
25 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
61 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
44 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
24 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
33 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
84 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
159 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
58 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
1answer
183 views

$\Delta$-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
39 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
40 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
46 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
34 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 ...