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

learn more… | top users | synonyms (1)

0
votes
0answers
15 views

Orthogonal Array Planes

In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ...
2
votes
1answer
127 views

nice space with wild fundamental group

I would like to know an example of nice space with very strange fundamental group. With simplices and similar things I only get finitely presented groups. Edit. I know from comments that Hawaiian ...
1
vote
1answer
62 views

Computing homology group of product of spheres

I am having trouble computing $\tilde{H}_n(S^3\times S^1)$. I am supposed to use Mayer-Vietoris sequence. I know $\tilde{H}_n(A\vee B)\cong\tilde{H}_nA\oplus\tilde{H}_nB$ if there is a contractible ...
0
votes
0answers
29 views

is there a specific way to find deck transformation and its related group?

is there a specific way to find deck transformation and its related group? this question came to my mind at the first time I studied deck transformation and related topics of covering space. it ...
2
votes
1answer
61 views

Constructing a ring in Top by wedging and smashing pointed spaces

I'll list some things I believe I can do, any of which might I might not actually be able to, and then, if I'm right, I'll ask if there's any point to what I've done. The category of pointed locally ...
0
votes
1answer
52 views

A certain homotopy equivalence…

A few friends and I have been stuck on this old qualifying question for quite some time now... Let $D$ be the diagonal subspace of $\Bbb S^2 \times \Bbb S^2$. Show that the projection onto the first ...
0
votes
1answer
67 views

Induced nontrivial homomorphism?

I am caught up in a minor detail on a qualifying exam problem I am doing: Show that there are no injective continuous functions $f: \Bbb R^n \to \Bbb R^2$ , $n>2$ with $f(0)=0$. So far I have ...
1
vote
1answer
33 views

Is there a name for the result that homotopy groups are insensitive to removing submanifolds of sufficiently large codimension?

First some motivation. Consider $\mathbb{R}^n-\{0\}$. This is simply connected iff $n > 2$, since it deformation retracts to $S^{n-1}$. If instead we consider $\mathbb{R}^n - L$ where $L$ is a ...
2
votes
3answers
181 views

Is it possible a trivial fiber bundle with nonzero holonomy?

Let $P\rightarrow M$ be a principal bundle with structure group $G$. Suppose that the bundle is trivial $M\times G$; is it possible to have a nonzero holonomy along some closed trajectory on $M$ for ...
0
votes
1answer
96 views

A basic question on Relative Homology

And so this week, our algebraic topology class starts with relative homology groups. But there are some (REALLY) basic parts of the definition of the relative homology group that I don't understand ...
2
votes
3answers
82 views

Why is the fibre of each point compact?

For a compact covering space, the fibres of the covering map are finite. I am working on the same question as the one posed in this link, but there was an unanswered question at the end, namely, why ...
0
votes
1answer
14 views

Degree of the map and path components

I am trying to solve the following question: for each $a\in \mathbb{C}\setminus S^1$ calculate the degree of the map $$ \phi_a: S^1\to S^1; z\mapsto \frac{z-a}{|z-a|} $$ and deduce that $a,b\in ...
2
votes
1answer
94 views

Question about the definition of homology

$\quad$ The functor $H_n$ measures the number of “$n$-dimensional holes” in the space (or simplicial complex), in the sense that the $n$-sphere $S^n$ has exactly one $n$-dimensional hole and no ...
3
votes
2answers
153 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
0
votes
0answers
29 views

branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
1
vote
1answer
58 views

Tangent bundle of a manifold [duplicate]

can anyone help me with this problem: Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$? for the 1st ...
0
votes
0answers
20 views

why $i_{1_{*}}(r)=aba^{-1}b^{-1}$?

my question is about this example,I just didn't understand why $i_{1_{*}}(r)=aba^{-1}b^{-1}$? I want to visualize how the generator $r$ changed to $aba^{-1}b^{-1}$ in $S^{1} \vee S^{1}$ but I ...
1
vote
1answer
90 views

Construction of Hodge decomposition

We know Hodge decomposition splits any $k$-form into three $L^2$ components. And I see some proofs, none of them provide an explicit constructive method. Is there any general method to construct one? ...
0
votes
1answer
67 views

$X \simeq Y \Rightarrow \pi_0(x) \cong \pi_0(y)$

Let $X$ and $Y$ be topological spaces. $X \simeq Y \Rightarrow \pi_0(X) \cong \pi_0(Y)$ I'm trying to prove this and I have no idea where to begin. Any hints would be helpful but I really don't want ...
1
vote
2answers
46 views

Relations between the homology group of quotient space and the relative homology group

Let $X$ topological space with subspace $A$.Under what conditions,$H_{n}(X,A)$ is isomorphic to $H_{n}(X/A)$
0
votes
2answers
50 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
8
votes
1answer
87 views

Equality of rank for homology and cohomology groups via the universal coefficient theorem

I'm having trouble understanding a passage from the proof of Corollary 3.37 in Hatcher's Algebraic Topology, namely the fact that the universal coefficient theorem implies $$ ...
7
votes
1answer
145 views

Uniqueness of curve of minimal length in a closed $X\subset \mathbb R^2$

Suppose $X$ is a simply connected closed subset of $\mathbb R^2$. Let $a,b$ belong to $X$. Is it true that there is at most one curve inside $X$ from $a$ to $b$ such that the length of the curve is ...
0
votes
1answer
31 views

Are homotopy types of finite CW complexes countable?

I need help with solving this problem. I came across this in a paper called "Counting homotopy types of manifolds".
3
votes
1answer
72 views

Questions on “simple-connectedness-like” property

I wanted to know if there's any notion which is very similar to the simple connectedness, but defined "purely" in terms of points and sets. Here's my attempt to do it. Let $X$ be a topological space. ...
0
votes
1answer
54 views

Homotopy equivalences

If $x\in X$, let $C(x)$ the path component of $x$ (the biggest path connected set containing $x$), and similarly if $y\in Y$. Let $C(X)$ and $C(Y)$ the family of all path components of $X$ and $Y$. ...
1
vote
1answer
49 views

If $X$ is homeomorphic to $Y$, does every map $X \to X$ factor through a map $Y \to Y$?

Let $X$ and $Y$ be topological spaces, and $h:X \to Y$ a homeomorphism. For every continuous map $f:X \to X$, is there a continuous map $g:Y \to Y$ such that $f=h^{-1} \circ g \circ h$? This came up ...
1
vote
1answer
26 views

Understanding an Induced Product on $\pi(G)$ Distinct from the Usual Product on $\pi(G)$

Hypothesis: Let $G$ be a topological group. Let $\mu$ denote the group multiplication mapping on $G \times G$ to $G$ that is stipulated to be continuous. Let $\pi_1(G,e) = \pi(G)$ denote the ...
0
votes
2answers
58 views

What surface is represented by $a_1 a_2 \cdot \ldots \cdot a_n a_1^{-1} a_2^{-1} \cdot \ldots \cdot a_n^{-1}$?

Question: What surface is represented by $a_1 a_2 \cdot \ldots \cdot a_n a_1^{-1} a_2^{-1} \cdot \ldots \cdot a_n^{-1}$? Attempt: I'm not sure where to even begin except I can make a few of the ...
3
votes
1answer
66 views

Showing that $\left\langle a,b \mid abab^{-1}\right\rangle \cong \pi_1(K) \cong \left\langle c,d \mid c^2 d^2 \right\rangle$

Hypothesis: Let $G$ and $H$ be defined in terms of the following presentations: $$ G \cong \left\langle a,b \mid abab^{-1}\right\rangle $$ $$ H \cong \left\langle c,d \mid c^2 d^2 \right\rangle $$ ...
1
vote
1answer
44 views

Existence of a suitable cover for $S^{2}$ and a given sheaf

I am trying to find a Leray covering for the 2-sphere with respect to the sheaf $\mathcal{F}=\mathbb{Z}$. I am also assuming that a contractible open covering satisfies $H^{i}(U,\mathbb{Z})$ for all ...
5
votes
1answer
44 views

Applications of Brown's Representability Theorem

I am currently trying to understand the proof of Brown's Representability Theorem, which says that any generalized cohomology theory is represented by an $\Omega$-spectrum. Can anyone point me to some ...
2
votes
0answers
52 views

Mapping cone z to z^2 on S1 is RP2

I want to show that the mapping cone of $z \mapsto z^2$ on $S^{1}$ is homeomorphic to $\mathbb{RP}^{2}$. My thought was: 1) $\mathbb{RP}^{2}$ is indeed $S^{2}$ with identifying each pair of ...
0
votes
0answers
35 views

Cohomology of Circle from unreduced Eilenberg-Steenrod Axioms

I would like to compute the cohomology groups of $S^1$ straight from the unreduced Eilenberg-Steenrod axioms. My motivation is to be able to calculate the cohomology group of spheres in any dimension ...
1
vote
0answers
36 views

Open Book Decompositions of 3-manifold and Associated Heegard Splittings

In page 13 of the paper: http://arxiv.org/pdf/math/0510639v1.pdf It is stated that "An open book decomposition (S,h, K) , gives rise to a special Heegard decomposition of M ". Here, S is a surface , ...
1
vote
2answers
57 views

Proving the existence of some deformation retract

I am trying to find out if the set $ S ^n \times S^n \setminus \left\lbrace (x,x) \mid x \in S^n \right\rbrace$ deformation retracts onto the subspace $\left\lbrace (x,-x) \mid x \in S^n ...
2
votes
1answer
81 views

equivariant cohomology in case of free actions (basic question)

Suppose $X$ is a topological space and $G$ is a topological group, and $G$ acts on $X$. Here is my question: If $G$ acts freely on $X$, then what are the maps showing $(X \times EG)/G$ is homotopy ...
3
votes
0answers
55 views

Can injective and projective model structures be the same ?

Is there a non trivial example where injective and projective model structures coincide ? By non trivial I mean an example of a functor category on a non discrete base category where I guess ...
3
votes
1answer
79 views

Question about Relative Cohomology

I need help with the following question please: Suppose that a space $X \subseteq Y $ retracts onto some subspace $A \subseteq X $. When do I have $H^\ast ( Y,X) \cong H^\ast (Y,A)$? Thanks.
0
votes
1answer
27 views

The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices

"The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices." Is this always true? The barycentric centers are equidistant from the vertices. k=2 So ...
1
vote
1answer
35 views

Identifying letters up to homotopy

I already identified the letters of the alphabet up to homeomorphism and the useful characteristic was cut-points and their preservation under homeomorphism. As a visual representation you can imagine ...
0
votes
1answer
43 views

Is a Covering Space of a Topological Space always Hausdorff?

Is a Covering Space of a Topological Space always Hausdorff? I can separate two different points from the same fiber, but what about two arbitrary points?
2
votes
2answers
182 views

A more general definition of branched covering.

If $f:X\longrightarrow Y$ is a holomorphic map between two compact Riemann surfaces, then $f$ is called also a branched covering map. This because the branched points of $f$ form a finite set ...
2
votes
1answer
123 views

Solving an exercise in Milnor-stasheff's “characteristic classes”

I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let $ M =S^n $ be the $n$-sphere and let $TM$ be its tangent ...
10
votes
1answer
130 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
2
votes
1answer
123 views

Refining homotopy commutative maps of spectra to maps of E_{\infty}-ring spectra

In Adams' blue book (page 54) we have a map in the homotopy category of ring spectra $f: MU \rightarrow K$ where $K$ is complex $K$-theory such that $g_*x^{MU} = (u^K)^{-1}x^K$ where $x^E$ denote ...
6
votes
1answer
88 views

De Rham cohomology of $T^*\mathbb{CP}^n$

I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of $T^*\mathbb{CP}^n$ (seen as a real manifold). Now, this should be equal to ...
1
vote
1answer
48 views

Can we compare cohomology rings with different coefficients?

I have an example sheet that asks me to compute the cohomology rings for two spaces, say X and Y, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_d$ respectively. It then asks whether X and Y are ...
1
vote
1answer
50 views

Poincare lemma on current

The Poincare lemma on current states that: If $U$ is a star-shaped open set in $\mathbb R^n$ and $T$ is a $k$-current on $U$ such that $dT=0$, then there is a $k-1$-current $S$ on $U$ such that $dS = ...
2
votes
1answer
57 views

$S^{1}$-bundles over $\mathbb{RP}^2$

How many $S^1$-bundles over $\mathbb{RP}^2$ do exist? Is it true that there exist only two bundles - trivial and not?