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

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9
votes
1answer
535 views

Showing that a zigzag space is contractible

I'm trying to solve part (b) of exercise 0.6 in Hatcher's Algebraic Topology: (b) Let $Y$ be the subspace of $\mathbb{R}^2$ that is the union of an infinite number of copies of $X$ arranged as in ...
4
votes
1answer
66 views

Injectivity between non-trivial knot on torus and $S^1$ on torus.

$X = S^1 \times D^2$ and $A$ the circle shown in the figure, Show that there are no retractions $r \colon X \to A$. Assume for contradiction that there is a retraction $r \colon X \to A$, then ...
1
vote
1answer
113 views

Hachter 1.1.14, prove on isomorphism by projection.

I am a bit confused with what I am suppose to prove here. I plan to go with prove isomorphism = homomorphism + bijection, but which function should I construct for the homomorphism? Show that the ...
0
votes
0answers
20 views

A question about how we get from PreSheaves of X to those of Y - why is $f_{*}F(V) = F(f^{t}(V)) \in PSh(k_Y)$

The problem that I am struggling with is the meanings and relationships of $f_{*}, f^{t}, f^{-1}, \text{and} f^{\dagger}$. These definitions come from Shapirra's notes on Algebra and Topology which ...
2
votes
2answers
225 views

$A$ retract of $X$ and $X$ contractible implies $A$ contractible.

I have constructed the following proof of the statement and have some questions (a question) about the correctness of the proof: Statement: $A$ retract of $X$ and $X$ contractible implies $A$ ...
3
votes
0answers
132 views

(Topological quantum field theory) identifying objects of cobordism category

I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I ...
4
votes
2answers
115 views

Cohomology and Global Sections

For a topological space X, $ \ H^0 (X, \Bbb Z)$ tells you about the connected components of $X$. For a sheaf $\mathcal O_X$ on $X$, $H^0 (X, \mathcal O_X)$ is usually written to refer to global ...
15
votes
1answer
381 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
1
vote
1answer
122 views

What is the meaning of concircular..

What is the meaning of the concircular? It says that the any five points are pairwise nonparallel and no four of them are concircular.
3
votes
2answers
178 views

Generalization of the hairy ball theorem.

The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Can the hairy ball theorem be strengthened to say that there is no ...
1
vote
2answers
1k views

Difference between Deformation Retraction and Retraction

I am currently reading through Hatcher's Algebraic Topology book. I am having some trouble understanding the difference between a deformation retraction and just a retraction. Hatcher defines them as ...
1
vote
2answers
78 views

Changing the direction of the homomorphism in the definition of pushouts

In the diagram below $P$ is a pushout of the data if a unique $u:P\rightarrow Q$ exists for every solution $(Q,j_1,j_2)$ (by the definition of bushouts). My question is : Suppose that the definition ...
5
votes
2answers
267 views

Who proved that existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ was sufficient for HEP?

It is well known that the existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is necessary to make $\left(X,A\right)$ a pair having the homotopy ...
1
vote
1answer
57 views

intersection form for non-compact manifold

Can we define the intersection form $H_{k}(M) \times H_{n-k}(M) \rightarrow \mathbb{Z}$ for non-compact manifold $M^n$?
2
votes
1answer
87 views

Calculating the homotopy groups of a complex

I'm trying to compute the homotopy groups of the complex obtained by gluing two Klein bottles along the generator that preserves orientation. It's not dificult to compute the fundamental group, ...
0
votes
2answers
32 views

If $\mathcal{C}$ is the category with one object, then a natural transformation from $\mathcal{C}$ to $\mathcal{D}$ is a morphism in $\mathcal{D}$

I'm not sure what suffices as a proof of the above statement. We know that a functor from $\mathcal{C}$ to $\mathcal{D}$ is equivalent to a choice of an object in $\mathcal{D}$, so this statement ...
10
votes
3answers
440 views

Statement about Homotopy in Brown's “Topology & Groupoids”

I am trying to understand a statement in Brown's Topology and Groupoids, 7.2.5 (Corollary 1), page 270. Let's first have some preliminary remarks Let $X,Y$ be topological spaces. The track groupoid ...
3
votes
2answers
261 views

Show that $X$ deformation retracts to any point in the segment $[0,1]\times \lbrace 0 \rbrace$, but not to any other point.

I'm trying to solve a problem from Hatchers "Algebraic Topology" - exercise 0.6 (a): Let $X$ be the subspace of $\mathbb{R}^{2}$ consisting of horizontal segment $[0,1]\times \lbrace 0 \rbrace$ ...
12
votes
2answers
210 views

If a smooth manifold X is covered by an odd sphere, then X is orientable.

In solving some old qualifying exam questions, I've been thoroughly stumped. If a smooth manifold $X$ is covered by an odd dimensional sphere, then $X$ is orientable. I see this question has ...
1
vote
1answer
53 views

Constructing a chain complex to induce an isomorphism on homology

Say we have the $\mathbb{Z}/n\mathbb{Z}$ for $n \geq 2$ as a chain complex concentrated at degree zero, i.e. $\cdots \rightarrow 0 \rightarrow \mathbb{Z}/n\mathbb{Z} \rightarrow 0 \rightarrow ...
0
votes
2answers
298 views

Is a path connected subspace of a simply connected space simply connected?

This is sort of a lemma I'm trying to prove for a larger proof. It seems intuitively true: if a space has trivial fundamental group, any two loops based at a point are homotopic. A subspace of such a ...
1
vote
1answer
50 views

Identity in Thom spaces.

Let $T$ be the one-point compattification, $E$ a real vector bundle, $\epsilon$ the trivial line bundle and $\Sigma$ the suspension operation. How can I prove that $$ T(\epsilon \oplus E) \simeq ...
1
vote
0answers
37 views

Group of continuous automorphisms of cylindrical plane is a lie group

How do I prove this theorem? Theorem: The group of continuous automorphisms of a cylindrical plane is a lie group. In this context, cylindrical means the Laguerre plane. I found a paper it ...
1
vote
1answer
64 views

Fixing an object $Y$ defines a contravariant functor $\mathcal{C}^{op} \rightarrow$ Sets

Say we fix an object $Y$, and then consider the assignment $(\mathcal{C}^{op})_0 \rightarrow$ Sets that sends $X$ to $\hom_{\mathcal{C}}(X, Y)$. How can I show that this defines a contravariant ...
0
votes
1answer
122 views

Virtual dimesion and index?

I was told that the virtual dimension (not sure if it is virtual cohomology dimension) is equal to the index of certain operators on a manifold. Does anyone know more about this possible relation? ...
0
votes
2answers
1k views

Deformation retract and homotopy equivalence

If $A\subset X$ is a deformation retract of $X$. Are $X$ and $A$ homotopy equivalent?
3
votes
1answer
164 views

Rational spectra

I keep reading the term "rational (ring-)spectrum" but can't find a definition. My original motivation for researching this was to understand some basic examples of complex oriented cohomology ...
2
votes
1answer
188 views

Klein bottle is the connected sum of two $\Bbb{R}P^2$'s

Question How do I show that the Klein bottle is the connected sum of two $\Bbb{R}P^2$'s? More precisely; how do I construct an explicit continuous bijection between the Möbius strip and ...
1
vote
2answers
109 views

Homology and cofibrations

Please help me to answer my question.It is finding $H_*(T^2,\lbrace\ast\rbrace\times S^1\cup S^1\times\lbrace\ast\rbrace)$ using cofibrations matter.I would be grateful for your answer.
0
votes
1answer
40 views

Describing the Tychonoff topology

my question is: Describe the Tychonoff topology on $Y^X$ in a manner similar to the description in below proposition of the compact-open topology. Proposition: If $X$ is locally compact ...
2
votes
2answers
948 views

Homology groups of the Klein bottle

I've seen this but didn't really understand the answer. So here is what I tried: According to this picture we have one 0-simplex - $[v]$, two 1-simplices - $[v,v]_a,[v,v]_b$ and two 2-simplices - ...
1
vote
1answer
154 views

induced map between cohomology groups of real projective spaces

Let $i: \mathbb{R}P^{n-1}\hookrightarrow \mathbb{R}P^n$. Then $i$ induces isomorphisms $i^*: H^k(\mathbb{R}P^n;\mathbb{Z}_2)\longrightarrow H^k(\mathbb{R}P^{n-1};\mathbb{Z}_2)$ for $0\leq k\leq n-2$. ...
2
votes
0answers
57 views

Signature form of $S^2 \times S^2$

Let $M=S^2 \times S^2$ be the product of two copies of the $2$-sphere. We have that $dim(M)=4$. So we can define the intersection form $$ I_{S^2 \times S^2} := H^2(M, \mathbb{Z}) \times H^2(M, ...
3
votes
0answers
80 views

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...
3
votes
1answer
65 views

Is there a natural example of a $K(\hat{\mathbf Z}, 1)$?

Does there exist a nice classifying space for $\hat{\mathbf Z}$, the profinite completion of $\hat{\mathbf Z}$?
7
votes
1answer
135 views

$H_0(X)\simeq\Bbb{Z}^k$, where $k$ is the number of path components

I want to prove that $H_0(X)\simeq\Bbb{Z}^k$, where $k$ is the number of path components of $X$. What I tried Since $∂_0=0$, ...
1
vote
1answer
66 views

What is the topological equivalence of a genus 2 torus?

Am I get really stuck, or it is not obvious: what is the topological equivalence of a genus 2 torus? We know torus is $S^1 \times S^1$. Then how about genus 2? Thank you~
1
vote
2answers
150 views

Covering map + homotopy equivalence = homeomorphism?

How to show that a covering map which is also a homotopy equivalence is a homeomorphism?
8
votes
2answers
282 views

If $\|\left(f'(x)\right)^{-1}\|\le 1 \Longrightarrow$ $f$ is an diffeomorphism

Let $f:\mathbb{R}^n \longrightarrow \mathbb{R}^n,f\in C^1(\mathbb{R}^n)$ such that $\forall x \in \mathbb{R}^n\;,\;f'(x)$ is an isomorphism and: $$ \|\left(f'(x)\right)^{-1}\|\le 1\;,\forall x \in ...
2
votes
0answers
39 views

Which perfect groups occur as fundamental groups of finite spaces?

Is there a way of constructing the Hasse diagram of a finite T0-space for a given perfect finite group?
1
vote
1answer
129 views

A function “extends” to the cone on X

I have the following statement: A map $f : X \rightarrow Y$ is nullhomotopic if and only if it extends to the cone on $X.$ My problem is that I have no idea what "extends" means in this statement (I ...
5
votes
2answers
127 views

Why is $D^{n+1}/S^{n} = S^{n+1}$ true?

I went to my first lecture in Algebraic Topology and managed to get really confused. It seems like they assumed that the following statement was "obvious": $D^{n+1}/S^{n} = S^{n+1}$ Where $D^{n}$ is ...
3
votes
1answer
183 views

Deformation Retraction to a point

I have a question from Hatcher's Algebraic topology Chapter 0, problem 6: "Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the horizontal segment $[0,1]\times\{0\}$ together with the vertical ...
3
votes
1answer
54 views

The smooth deformation

$M$ is a connected smooth manifold and $p$ is a fixed point on $M$. For a null-homotopic smooth loop $\gamma$ at $p$, can we find a smooth deformation, that is, a smooth function $f :[0,1] \times ...
3
votes
0answers
102 views

Subset of $\mathbb{R}^3$ with an element of finite order in its fundamental group

Is there a subset of $\mathbb{R}^3$ with an element of finite order (not the identity!) in its fundamental group? I think the real projective plane is such a subset as its fundamental group is ...
-1
votes
1answer
98 views

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative d is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative $d$ is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying: $d(ω + η) = dω + dη$; $ω ∈ Ω^p(U), η ∈ Ω^q(U) ⇒ d(ω ∧ η) = dω ∧ η + ...
2
votes
1answer
91 views

”figure 8” space embedded in S2

Let M3 be the 3-manifold defined as the quotient space of I × S2 by the identification {0} × {x} s {1} × {Tx}, where T : S2 → S2 is a reflection through a plane in R3. Find π1(M) and π2(M).
1
vote
2answers
234 views

a closed 1-form which is not exact

On the unit circle S1 in the plane, let θ = arctan(y/x) be the usual polar coordinate. Show that dθ makes sense on S1 and is a closed 1-form which is not exact.
3
votes
1answer
132 views

On continuously uniquely geodesic space II

This question was inspired by this answer of @wspin. Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints. Question ...
3
votes
2answers
175 views

top cohomology of a manifold with boundary

As stated in the title, my question is the following: Let $M$ be a compact orientable manifold with boundary $\partial M$. Is it true that $H_n(M;\mathbb{R})$ is always zero? In the trivial case ...