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

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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 ...
9
votes
1answer
299 views

What is known about $\operatorname{Aut}(\mathbb{I}^n)$

A few months ago, I asked a related question: Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$? It was interesting for me to know that ...
8
votes
1answer
277 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
7
votes
1answer
99 views

Homology of real projective space… I'm not satisfied with the argument in hatcher.

In example 2.42 Hatcher computes the homology of real projective space. I follow his argument, but I would be uncomfortable believing the details of the degree computation if I didn't see it in his ...
7
votes
1answer
110 views

Cohomology with coefficients in a commutative ring, how are the chain groups defined?

I have been studying a course in algebraic topology that follows Hatcher's textbook on the subject. I have some queries as to how certain things are defined. The first part of the text defines the ...
6
votes
1answer
111 views

Airy differential equation and Galois group

Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over ...
6
votes
1answer
129 views

A topological example from Church's undecidability paper

A. Church, in his classical paper An unsolvable problem in elementary number theory in American Journal of Mathematics Vol. 58 No. 2. (1936), pp. 345-363, (available here), wrote: There is a class ...
5
votes
1answer
125 views

Yoneda-type lemma for compositions on the hom-functor

The Yoneda lemma basically rephrases a rigidity property of natural transformations out of a covariant hom functor: A natural transformation $\psi : \mathsf{Hom}(Z,-)\Rightarrow F$ is determined by ...
5
votes
1answer
245 views

Homology groups of orientable surfaces.

Edit: I have a proof here but when I spoke last with my professor, she told me something was close, but not quite. Can someone help me patch this proof? I've been trying to get this down for quite a ...
5
votes
1answer
56 views

Module structure in the Serre spectral sequence of the Borel construction

Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its ...
5
votes
1answer
162 views

If $i\colon A\to X$ is a cofibration then $1\times i\colon B\times A\to B\times X$ is a cofibration for any space $B$. Is that true?

In Algebraic Topology (Hatcher, pg 14) I find: A pair $\left(X,A\right)$ has the homotopy extension property if and only if $X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is a retract of ...
5
votes
1answer
98 views

Bijection abstract simplicial complex

Given two compact Hausdorff spaces $X$ and $Y$ and $h \colon X \to Y$ a homeomorphism, how can I prove that $h_{\mathfrak{A}} : N(\mathfrak{A}) \to N(h(\mathfrak{A}))$ is a bijection where ...
5
votes
1answer
161 views

Homotopic maps in a directed system induce homotopic maps on colimit?

Let $(A_i,f_i)$ be a directed system of CW-complexes with colimit $A$. Further, let $g_i:A_i\to A_{i+1}$ be maps such that $g_{i+1}f_i=f_{i+1}g_i$ and $f_i\simeq g_i$. This might or might not be the ...
5
votes
1answer
104 views

About twistor space of a K3 surface

I know that for $X=(M,I)$ , where $I$ is the complex structure, a K3 surfaces and $\alpha \in H^2(X,\mathbb{R})$ a Kähler class, there exist a Kähler metric g and J,K complex structures such that 1) ...
4
votes
1answer
158 views

A question on Hawaiian earring

Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group ...
4
votes
1answer
38 views

Applications of the fact deformation retracts are closed under pushout

Fact: Suppose we have a pushout diagram $$\require{AMScd} \begin{CD} A @>{f}>> C\\ @V{i}VV @VV{j}V\\ B @>>{g}> D\end{CD}$$ where $i$ is the inclusion of a deformation retract. ...
0
votes
0answers
65 views

Prove on $S^1$ deg(f)=deg(g)=>f is homoptopic to g

Let $a(t)=e^{2\pi it}$ be the generator of $\pi_1(S^1,1)$ and we define degree of $f$ this way: $[\omega^{-1}]\cdot f_*([a])\cdot [\omega]=\deg(f)[a]$ where $\omega$ is any path from $f(a)$ to $1$. ...
0
votes
0answers
124 views

Klein Bottle considered to be two Möbius band $A,B$ glued together

In Hatcher, a Klein Bottle $K$ is considered to be two Möbius band $A,B$ glued together. I see the map $\phi$ is $\mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$, but why $1 \mapsto (2,-2)$? I other ...
0
votes
0answers
52 views

Seemingly serious problem with Deformation Retraction

So a friend and I are arguing over Deformation retraction. Any help to settle this would be nice. Consider a T shaped subspace of $\mathbb{R}^2$. Let $A$ be the vertical segment and let $B$ be the ...
0
votes
0answers
38 views

fibre bundle on [a,b]. prove that every fibre bundle on it is trivial.

prove that every fiber bundle on [a,b] is trivial. Please prove this elaborately. Is the Lebesgue covering lemma is required to prove this result?
0
votes
0answers
150 views

acyclic implies identity null-homotopic?

I have proved the following for a chain complex $\mathcal{C}_{*}$ where the $\mathcal{C}_i$ are free $\mathbb{Z}$ modules, $\mathcal{C}_i = 0$ for $i>0$. The identity map on $\mathcal{C}_{*}$ is ...
0
votes
0answers
42 views

Intuition behind a braid operator which is also a solution for Yang-Baxter equation

I am going through this paper, 'Quantum entanglement and topological entanglement' by Louis H Kauffman and Samuel J Lomonaco Jr published in New Journal of Physics 4 (2002). It started with ...
0
votes
0answers
47 views

Constant sheaves associated with locally closed subsets

I'm studying P. Schapira's notes Algebra and Topology, available online here, and I'm having trouble understanding sheaves associated with locally closed subsets, in particular constant sheaves. For ...
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 ...
0
votes
0answers
169 views

Is the interior of simply closed curve homeomorphic to disk?

I am strongly guessing that the interior of any simple closed curve in $R^2$ should be homeomorphic to unit disc $D$. But I cannot prove it. Is it true? Then can you shed me some idea for proving ...
0
votes
0answers
68 views

Is a uniquely geodesic space contractible? II

Is a uniquely geodesic space, contractible ? With the extra assumption that closed metric balls are compact, there is an answer here. We expect here an answer beyond this extra assumption ...
0
votes
0answers
134 views

What's the classification of CW complexes formed by gluing a 2-cell to a circle?

After this answer, the following question comes : What's the classification (up to homeo.) of CW complexes formed by gluing a 2-cell to a circle ?
0
votes
0answers
41 views

Path fibration over a connected manifold.

Let $M$ be a differentiable manifold. We can consider $P(M):=\{\gamma:[0,1] \to M\}$, so we have a natural projection on $M$ $$ P(M) \to M $$ $$ \gamma \mapsto \gamma(1) ,$$ in the fibre of this ...
0
votes
0answers
80 views

Clarification needed - the fundamental group of the circle

I am reading the proof of $\mathbb{Z}\approx\pi_1(S^1)$ from Hatcher and didn't understand the last paragraph in the picture (the homomorphism part): Isn't $\tau_m\tilde{\omega}_n:I\to \Bbb{R}$ ...
0
votes
0answers
110 views

Showing that $S^2\times S^3$ is not homotopic to $S^2 \vee S^4 \vee S^6$

In Switzer's algebraic topology book on page 285 he shows that $X = S^2 \times S^3$ is not homotopic to $Y = S^2 \vee S^4 \vee S^6$ by showing their cohomology rings are different. In doing so, he ...
0
votes
0answers
36 views

Multiplication by an integer on a symmetric spectrum

The map $n : X \rightarrow X$ is used in the definition of the mod-n spectrum $X/n$. But what is this map? How does it look like? I know how this works for the sphere spectrum (just use the map $z ...
0
votes
0answers
81 views

Question about an isomorphism result in homotopy theory

I have another question regarding homotopy theory and winding numbers (or degrees). In Manton and Sutcliffe they state the following theorem: $\pi_2(G/H)=\pi_1(H)$ provided $G$ is a compact, ...
0
votes
0answers
91 views

Application of Kunneth formula to chain maps (Hatcher exercise)

I'm working on the following problem from Hatcher, which is in the Kunneth Formula section at the end of the cohomology chapter, and I'm having trouble figuring out where to start. Any direction would ...
0
votes
0answers
175 views

Every compact, connected, orientable surface without boundary admits a $\Delta$ complex structure

Every compact, connected, orientable surface without boundary admits a $\Delta$ complex structure I'm not sure if the approach I'm using is even correct. But basically, I know that every compact ...
0
votes
0answers
64 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
0
votes
0answers
83 views

Topolologics $(2n+1)$-dimensional Manifold with ball removed orientable?

Suppose you have a compact, orientable $(2n+1)$-manifold $M$. You take a neighbourhood about a point homeomorphic to $\mathbb{R}^{2n+1}$, and remove a small ball $B$. So the the boundary of ...
0
votes
0answers
80 views

A Question on Algebraic Topology

I am reading algebraic topology by Artin, and he gives the following definition. Let $X$ be a topological space. We define a homomorphism $sd_X: S_p(X) \to S_p(X)$ by induction. If $T: \Delta_0 \to ...
0
votes
0answers
88 views

Algebraic Topology chain maps

I came across the following statement in a textbook I am reading, and I was wondering what is meant by the abelian group of homotopy classes of chain maps. Could you help me out? Thanks
0
votes
0answers
135 views

Bott periodicity

I studied (using Morse theory) Bott periodicity theorem for the unitary group $U(n)$: $\pi_{k}(U)=\pi_{k+2}(U)$. Do you know some interesting application of this result? Can this theorem help you to ...
0
votes
0answers
50 views

Do “multiples” of open dense sets of an algebraic group union to the whole group?

Let $G$ be an algebraic group. From algebraic geometry we know there exists an open dense subset $U$ of $G$ such that $U$ is nonsingular. Since left multiplication of $U$ by elements of $G$ is an ...
0
votes
0answers
46 views

An variation of “Lk” in simplicial complexes

Let $\mathcal S$ be a simplicial complex. For a simplex $S \in \mathcal S$, we have the closure, the star and the link $\operatorname{Cl} S := \{ F \in \mathcal S : F \leq S \}$ $\operatorname{St} S ...
0
votes
0answers
67 views

Non commutativity of Hopf space

How do one prove that $ \Omega S^{2}$ and $\Omega (S^{2}\vee S^{2} )$ are non-commutative H space?
0
votes
0answers
42 views

Showing that equivalence of loops is preserved by the product of loops

I have tried to prove that the equivalence of loops is preserved by the $\ast$ product, i.e. if $\alpha \sim_{x_0} \alpha'$ and $\beta \sim_{x_0} \beta'$, then $\alpha \ast \beta \sim_{x_0} \alpha' ...
0
votes
0answers
252 views

Induced map on homology

If I'm given a map from $\mathbb{C}P^1\times\mathbb{C}P^1$ to $\mathbb{C}P^3$ which sends $([z_{0},z_{1}],[w_{0},w_{1}])$ to $[z_{0}w_{0},z_{0}w_{1},z_{1}w_{0},z_{1}w_{1}]$, how do I compute the ...
0
votes
0answers
118 views

How do I prove this surface is homeomorphic to the sphere

Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$. How can I prove that S is homeomorphic to the sphere? I'm ...
0
votes
0answers
96 views

G is a topological group acts on topological space $X$, is $f_{g}:X\rightarrow X, x\rightarrow g*x$ continuous?

Let $G$ be a topological group acts on the topological space $X$, for an elememt $g\in G$, let's define the map $f:X\rightarrow X, f(x)=g*x$. I am trying to find if $f$ is continuous? my best ...
0
votes
0answers
473 views

$\Delta$ complex structure for $S^n$.

For n=1. We consider two copies of $\Delta^1$, $1$ simplicies and identifying their boundaries we get a loop, that is $S^1$. For n=2, identifying boundaries of two copies of $\Delta^2$ via identify ...
0
votes
0answers
75 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 ...
0
votes
0answers
151 views

Fiber bundle has homotopy extension property

I'm trying to understand the last bit of the proof. But, it makes no sense as if we defined $G=(H(x), G(R(X))$ won't we get some crazy recursive function. Does he mean $G=(H(x),R(x))$. Also, is ...
0
votes
0answers
195 views

polynomial cohomology

Hope this finds you all well. I want to make sure of one thing : Do we usually have polynomial cohomology only in case the cohomology modules are free of rank 1 at most in each degree? PS:I don't ...