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

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5
votes
2answers
426 views

deformation retract of $GL_n^{+}(\mathbb{R})$

Well, I need a deformation retract from $GL_n^{+}(\mathbb{R})$ to $SO(n)$ Here is what I tried, let $A\in GL_n^{+}(\mathbb{R})$ $A=(A_1,\dots,A_n)$ where $A_i$'s are collumn vectors, Recall that the ...
0
votes
2answers
262 views

Covering space and Fundamental group

Let $p:E\to X$ be a covering space and $\pi_1(E)$ be a fundamental group of $E$. Can you give me a recept for calculating a fundamental group $\pi_1(X)$ (may be for some special cases)? Thanks a lot! ...
4
votes
1answer
473 views

fundamental group of $GL^{+}_n(\mathbb{R})$

I would like to know whether the $GL^{+}_n(\mathbb{R})$ the set of all invertible matrices with positive determinant is simply connected or not? I guess it is not simply connected but that is just a ...
5
votes
2answers
432 views

Understanding the cartesian product of complex projective lines.

I am trying to understand the space obtained by taking the cartesian product $\mathbb{C}\mathbb{P}^1\times \mathbb{C}\mathbb{P}^1$ and identifying some of its points by the rule $(x,y)\sim (y,x)$. ...
6
votes
1answer
219 views

Pointed cofibrations between well-pointed spaces

Recall that we call a map $i: A \rightarrow X$ a cofibration if it has the homotopy extension property. We will say a pointed space $X$ is well-pointed, if the inclusion of the basepoint $\{ * \} ...
4
votes
0answers
73 views

homotopy type of the closure of a subset

Let $X$ be a topological space and $N$ a subset of $X$. Is it true that the closure of $N$ in $X$ is homotopy equivalent to $N$. I think it is not. take for example $N=\mathbb Q\subset \mathbb ...
6
votes
4answers
2k views

Representation of $S^{3}$ as the union of two solid tori

Well, I'm trying to prove that you can express the 3-dimensional sphere $S^{3}$ as the union of two solid tori. I tried first use that a solid tori is homeomorphic to $S^{1}$$\times$$D^{2}$ and use ...
1
vote
1answer
128 views

Complement of deformation retract

Let $X$ be a topological space and $V$ and $N$ are subspaces of $X$ such that $N$ deformation retracts onto $V$. I want to show that $X-V$ deformation retracts onto $X-N$. So i need to construct a ...
2
votes
2answers
545 views

Retraction map from unit disk to its boundary

Given two continuous surjective functions $f$ and $g$ from the unit disk to itself and $f(z) \neq g(z)$ for all $z$ in the unit disk is it possible to construct a retraction map from the unit disk to ...
3
votes
1answer
117 views

What does it mean to say that a pair of points are antipodal in a topological sphere?

A pair of points are antipodal if they are diametrically opposite to each other. This definition makes perfect sense when one thinks of the unit 2-sphere centered at the origin and embedded in $R^3$; ...
0
votes
0answers
74 views

First examples in triangulations

I am starting to study about triangulations in my algebraic topology course. We have seen the triangulation of the sphere, the closed disc and so on. Intuitively it's ok, however I couldn't find any ...
7
votes
5answers
239 views

How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
17
votes
2answers
5k views

“A proof that algebraic topology can never have a non self-contradictory set of abelian groups” - Dr. Sheldon Cooper

In the current episode "The Big Bang Theory", Dr. Sheldon Cooper has a booklet titled "A proof that algebraic topology can never have a non self-contradictory set of abelian groups". I'm still an ...
1
vote
1answer
191 views

homotopy type of the manifold minus the boundary

Let $X$ be a topological manifold with boundary.What is the idea behind the fact that emoving the boundary doesn't change the homotopy type of the manifold; i.e.,that is the manifold $X$ has the same ...
4
votes
2answers
157 views

open subsets in topological groups

I'm starting to study topological groups, and I noticed that Every single theorem in topological groups I have to use the following statement: Let $G$ be a topological group and U an open subset of ...
4
votes
1answer
151 views

Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
2
votes
0answers
109 views

Multiplicativity of the Euler characteristic

One can find all over the internet that it is well-known (and obvious) that given a fiber bundle $F \to E \to B$, the equality $\chi(E) = \chi(F)\chi(B)$ holds ($\chi$ is the Euler characteristic). ...
3
votes
1answer
138 views

etale fundamental group of a product

Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism $\pi_1(X \times ...
1
vote
0answers
109 views

Definition of Cellular action

Let X be a G-space and an ordinary CW-complex.We say that G acts cellularly on X if the following holds: 1) For each g $\in G $ and each open cell E of X, the left translation gE is again open cell ...
8
votes
0answers
257 views

When are maps between topological manifolds automatically surjective?

Take an injective (continuous) map $T:\mathbb{S}_1\to\mathbb{S}_1$. It's an obvious fact (though I can only prove it with nontrivial facts about homotopy) that $T$ is automatically surjective. I have ...
4
votes
1answer
96 views

Gluing axiom of a TQFT

In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT. When they come to prove the gluing axiom, they just mention that "...This statement is ...
3
votes
1answer
94 views

What does this free quotient space look like?

Let $S^2=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2=1\}$ and $S^1=\{(s,t)\in \mathbb{R}^2|s^2+t^2=1\}$. Suppose that $\mathbb{Z}/2\mathbb{Z}$ acts on $S^2\times S^1$ in such a way that the generator of ...
2
votes
1answer
156 views

How do I prove that $\mathbb CP^n$ is a 2n-manifold?

I'm struggling to prove that $\mathbb CP^n$ is 2n-manifold. We can defined the $\mathbb CP^n$ as the equivalence relation $(z_1,z_1,...,z_{n+1})\sim(w_1,w_1,...,w_{n+1})$ iff $z_i=\lambda w_i$, ...
4
votes
1answer
98 views

Finding homotopy equivalence

This is part of a problem from Hatcher: Show that the space in $\mathbb R^2$ which is the union (for $n \in \mathbb N$) of circles $C_n$, where $C_n$ is the circle centered at $(n,0)$ with radius $n$ ...
0
votes
1answer
147 views

When is cone homeomorphic to cylinder

This is the problem: Find non-trivial example of (linaer connected) space X so that cone over X is homeomorphic to cylinder over X. Trivial examples are one point set and empty set. I have absolutely ...
1
vote
0answers
118 views

Universal cover as a principal $\pi_1$ bundle.

Let $M$ be a connected manifold with universal cover $\tilde M$ and fix $x_0 \in M$. Then it is well-known that $\tilde M \to M$ is a principal $\pi_1(M,x_0)$ bundle. I'm a bit confused about the ...
3
votes
1answer
489 views

homotopic between two maps imply the homotopy between their mapping cone

Recall the mapping cone of a map $f: X\rightarrow Y$ is defined as the space $C_f: X\times [0, 1]\dot{\cup} Y/\sim$, where $\sim$ is the equivalence relation given by $(x, 1)\sim f(x)$ and $(x, ...
5
votes
1answer
73 views

Representations of $\pi_1M$ and Heegaard Splittings

I am reading Floer's Instanton-Invariant paper, and am stuck on a sentence. To set the stage: Consider a closed connected oriented 3-manifold $M$ and the nonabelian group $SU_2$. Denote the ...
4
votes
1answer
129 views

Fundamental group of a knot

If the circle and any knot are homeomorphic as topological spaces, why do they have different fundamental groups?
2
votes
1answer
318 views

Existence of tubular neighborhood

Let $X$ be a topological space and $A$ a subset of $X$. My understanding of a tubular neighborhood $N$ of $A$ is that $N$ is an open set containing $A$ such that $\bar N$ is a manifold with boundary ...
1
vote
2answers
292 views

Subset of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$

Let $X$ be a subset of $\mathbb{R^n}$. The topology on $X$ is induced by the topology of $\mathbb{R^n}$. If there is an homeomorphism from $X$ onto $\mathbb{R}^n$, is it true that $X$ is open in ...
0
votes
2answers
365 views

Need clarification on Exercise 2.2.7 Hatcher

I am looking at Exercise 2.2.7 of Hatcher (pg 155): For an invertible linear transformation $f: \Bbb{R}^n \to \Bbb{R}^n$ show that the induced map on $H_n(\Bbb{R}^n,\Bbb{R}^n - \{0\}) \approx ...
1
vote
1answer
66 views

What is the equivariant cohomology of the 0-sphere acting on the 1-sphere?

View $S^1$ as the unit circle in the complex plane and let $S^0$ act by complex conjugation. What is the Borel equivariant cohomology $H^*_{S^0}(S^1;{\mathbb{Z}})$ of this action? I ask this question ...
3
votes
0answers
142 views

cohomology isomorphism

Let $X$ be a finite dimensional CW complex and $A$ be a closed subset in $X$ and $N$ a regular neighborhood of $A$ that deformation retracts onto it. why do we have for each $i$, $$H^{i}(X-A;\mathbb ...
2
votes
1answer
168 views

How can we prove that the North hemisphere is homeomorphic to RP²?

How we can prove that the North hemisphere is homeomorphic to the projective plane RP²?
2
votes
1answer
321 views

Collapsing the boundary of a Möbius Strip to a point

I'm strunggling to prove that when we collapse the boundary of a Möbius strip we obtain the RP² thanks
4
votes
2answers
558 views

Homotopy Question Help?

Let $X$ be a topological space and suppose $X_1$ and $X_2$ are spaces obtained by attaching an n-cell to $X$ via homotopic attaching maps. Show that $X_1$ and $X_2$ are homotopy equivalent. Proof: ...
2
votes
0answers
103 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
0
votes
1answer
334 views

reduced relative homology

In example 2.18 page 118 of Hatcher's Algebraic topology we read : Applying the long exact sequence of reduced homology groups to a pair $(X,x_0)$ with $x_0\in X$ yields isomorphisms ...
4
votes
2answers
275 views

concerning the definition of homotopy extension property

1) Let X be a topological space, and let A $\subset$ X. We say that the pair (X,A) has the homotopy extension property if, given a homotopy $f_t\colon A \rightarrow Y$ and a map $\tilde{f}_0\colon X ...
5
votes
3answers
2k views

Good textbook or lecture notes on Seiberg-Witten theory.

I am looking for a good introductory book for Seiberg-Witten theory. The only textbook I have now is Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds". ...
2
votes
2answers
2k views

Retraction of the Möbius strip to its boundary

Prove that there is no retraction (i.e. continuous function constant on the codomain) $r: M \rightarrow S^1 = \partial M$ where $M$ is the Möbius strip. I've tried to find a contradiction using $r_*$ ...
4
votes
1answer
147 views

Homotopy equivalence preserves semilocal simply connectedness

If $X$ and $Y$ are two homotopy equivalent space, and $X$ is semilocally simply connected (equivalently, $X$ has a universal cover) must $Y$ be semilocally simply connected? How would one prove/find a ...
5
votes
1answer
132 views

What is the easiest way to see $\langle \Sigma X, Y \rangle\cong \langle X,\Omega Y\rangle $

Let $X$ and $Y$ be topological spaces. Let $\langle X,Y\rangle$ denote the homotopy classes of maps from $X$ and $Y$. The reduced suspension $\Sigma(-)$ has the adjoint $\Omega(-)$. In other words, we ...
7
votes
2answers
359 views

Homotopic maps which aren't relative homotopic

Can anyone give an example of two continuous maps, let say $f,g \colon X \to Y$, such that the set $A =\{ x \in X \mid f(x) = g(x) \}$ is not empty, the two maps are homotopic but there's no ...
3
votes
0answers
72 views

Unramified functions between Riemann surfaces

Let $F:X\rightarrow Y$ a unramified holomorphic function between two compact Riemann surfaces. I don't understand why $F$ is a covering map. By a well-known theorem $F$ is surjective; then since the ...
19
votes
1answer
484 views

A homotopy sphere

My question is part of an exercise in Hatcher's 'Algebraic Topology'. Consider a CW complex $X$, constructed from a circle and two 2-disks $e_2$ and $e_3$, attached to that circle by maps of degree 2 ...
3
votes
2answers
499 views

Fundamental group of $SO(3)$

How can I show that the universal cover of $SO(n)$, for $n\ge 3$, is a double cover? And how does that reflect the fact that the fundamental group of $SO(n)$ has two elements? What is the relation ...
3
votes
1answer
589 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
1
vote
2answers
92 views

A basic question on relative singular groups\homology

I have some basic questions on relative singular homology. Let $A \subset X$ and let $C_k(X,A)$ be the relative singular k-chains. Then it is said that since $\partial$ carries both $C_k(X)$ and ...