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

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0
votes
1answer
27 views

Klein bottle and Real Projective plane

How to determine the triangulation of these two objects? can we use the above to compute Fundamental Group of Klein bottle and Real Projective plane? I can use the van kamen theorem to prove one is ...
0
votes
2answers
44 views

Computing fundamental groups

I want to prove that the fundamental group of the union of two spheres $S^m$ and $S^n$ joined to one point, and with $m,n\geq 2$ is trivial. I'm completely stuck so every help will be welcome. Thank ...
1
vote
1answer
32 views

Computing the homology groups of a quotient space of the sphere

I want to solve following question: Let $A$ denote the union of equatorial circle and the north pole on $S^2$. Let $X=S^2 / A$. Compute the homology groups of X. I calculated that $H_2(X) = \Bbb ...
2
votes
1answer
38 views

Some question about path connectedness

I think that's intuitively evident but I can't prove that the set $\mathbb{S}^n\setminus\{(0,\cdots, 1),(0,\cdots, -1)\}\; (n>1)$ is path connected. Does anyone have a formal argument to prove it? ...
0
votes
0answers
19 views

Regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $.

I tried to draw the regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $. I think the regular covering space is: Is it true? How do you draw the non-regular ...
-1
votes
1answer
16 views

Problem with Massey's exercise 3.3 [on hold]

I'm stuck with the following problem from Massey's book: "If $f,g$ are paths over $X$ with initial point $x_0$ and terminal point $x_1$ prove that $f$ is equivalent to $g$ iff $f\cdot \bar{g}$ is ...
1
vote
1answer
25 views

Pull-back of a fibration along a homotopy equivalence

Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ and $g:B\rightarrow B'$ be homotopy inverses. Denote by ...
1
vote
1answer
23 views

Is there any simply connected polyhedron with a not simply connected face?

According to Wikipedia, For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2. Is it really necessary to specify here, that ...
2
votes
0answers
52 views

Reduced homology groups of a space which is the union of finitely many open subsets

This is exercise 33 (p.158) from section 2.2 in Hatcher's Algebraic Topology: Suppose the space $X$ is the union of open sets $A_1, \ldots, A_n$ such that each intersection $A_{i_1} \cap \cdots ...
3
votes
0answers
20 views

Criterion for homotopy equivalence in the category of pair of spaces

I am trying to prove the following statement : Let $\{ * \} \subset A \subset B \subset X$ be a chain of topological spaces (all subsets have the subspace topology). It is given that $A \to X$ ...
0
votes
0answers
19 views

Is there a ''truncated'' long exact sequence of homotopy groups for a pullback of a ''truncated'' Kan fibration?

Let \begin{align} A &\to B\\ \downarrow &~~~~~\downarrow\\ C &\xrightarrow{f} D \end{align} be a pullback square of pointed simplicial sets. If $f$ was a Kan fibration, this would be a ...
0
votes
0answers
40 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...
2
votes
1answer
33 views

Tangent bundle of manifold with no odd dimensional sub-bundles

First, a preliminary remark: The Whitney sum of two vector bundles is orientable. I saw this statement somewhere and was wondering if it's true. In particular, it's easy to show that ...
2
votes
1answer
37 views

Approximating a CW-complex with a trivial fundamental group

Let $X$ be CW-complex and $\pi_1(X) = 0$. I want to prove that there is $Y \simeq X$ with only one $0$-cell and without $1$-cells. Geometrically it's obvious, but I have no idea for rigorous proof. I ...
2
votes
1answer
31 views

$\mathbb{R}P^4$ and $\mathbb{R}P^6$ do not admit fields of tangent $2$-planes

See the related question here. This is the second part of question 4-C in Milnor and Stasheff's book on characteristic classes. In the solution to the first part, we rely on the fact that having a ...
6
votes
2answers
83 views

Visualizing products of $CW$ complexes

I'm learning about products of CW complexes. The sources I've seen talk about the matter as follows: given topological spaces $X$ and $Y$ with a given CW decomposition, we can then form a CW ...
0
votes
2answers
47 views

How to compute a homotopy to show the operation on the fundamental group is assoicative?

By definition $$[(\alpha *\beta) *\gamma ] (s) = \begin{cases}\alpha (4s) & 0 \leq s\leq \frac{1}{4} \\ \beta(4s-1) & \frac{1}{4}\leq s\leq \frac{1}{2}\\ \gamma(2s-1) & \frac{1}{2}\leq ...
1
vote
2answers
57 views

Problem from Rotman's Algebraic Topology book

suppose $ n > m $ and $ i : RP^m \to RP^n $ is the natural imbedding.Then show that $ i^* : H^q(RP^n ; Z_2) \to H^q(RP^m ; Z_2)$ is an isomorphism for all $ q < m+1$ this is a problem from ...
0
votes
0answers
52 views

Adjunction between topological and simplicial presheaf categories

For a small discrete category $C$, the singularization/realization adjunction $Top \leftrightarrows sSet$ induces an adjunction $Top^C \leftrightarrows sSet^C$. If I have a small topological category ...
1
vote
1answer
50 views

If $ f : S^n \to R^n $ satisfies $f(-x)=-f(x)$ for all $x \in S^n $,then there exists $ y \in S^n $ with $f(y)=0 $

If $ f : S^n \to R^n $ satisfies $f(-x)=-f(x)$ for all $x \in S^n $,then there exists $ y \in S^n $ with $f(y)=0 $ This is a problem from Rotman's Algebraic Topology book. I think I have to use ...
2
votes
1answer
52 views

The Plate Trick and $SO(3)$

This is a followup of sorts to Plate trick demonstrating SO(3) not simply connected. . Suppose I do the ``plate trick'' to demonstrate the existence of an order-2 element in $\pi_1(SO(3))$. That is, ...
2
votes
1answer
63 views

An equivalence of categories

Let $F: \Pi(X) \to \text{SET}$ be a functor, where $\Pi(X)$ is the fundamental groupoid of $X$. I have shown earlier that we can construct a covering $p: Y = \bigsqcup_{x\in X} F(x) \to X$ from the ...
2
votes
1answer
58 views

What is meant by “constant” in Liouville's Theorem?

Liouville's Theorem states that: Every holomorphic* function for which there exists a positive number M such that $|f(z)| \le M$ for all $z \in \mathbb C$ is constant. I'm using this to prove ...
-3
votes
1answer
77 views

Can a torus have a simple map from two dimensions to three dimenions like a Gauss map? [on hold]

There are several ways to project two dimensions onto a Riemann Sphere and the Gauss map works very well. A Gauss map: 2d {x,y}-> 3d {2*x/(1 + x^2 + y^2), 2*y/(1 + x^2 + y^2), (1 - x^2 - y^2)/(1 + ...
2
votes
1answer
45 views

Poincaré lemma on a space with trivial homology group

Today I read about Poincaré's lemma from do Carmo's book Differential Forms and Applications. It says that A closed differential $k$-form on a contractible space is exact. I wonder if the ...
5
votes
1answer
63 views

Correct meaning of two spaces being homotopy equivalent under a space

Let $p_0 : A \to X_0 $ and $p_1 : A \to X_1$ be two maps. I am confused about what does it mean to say that '$X_0$ and $X_1$ are homotopy equivalent under $A$'. Which of the following statements is ...
3
votes
1answer
34 views

Fundamental question about cohomology on the stable homotopy category

I've gotten myself tied in knots about this elementary derivation of a ludicrous conclusion. Appreciate a hand straightening myself out! (1) A fibration $F\to E \to B$ of CW complexes gives rise to a ...
7
votes
3answers
265 views

Difference between Homology and Cohomology

Homology and cohomology are similar because the latter is the former acted by $\text{hom}$ functor, and we also have Theorem Let $C$ and $D$ be free chain complexes; let $\phi:C\to D$ be a chain ...
1
vote
0answers
12 views

Cohomological dimension of an orbit space in Alexander-Spanier theory

Take a ring $R$ and define $cd_R(X)$ ("cohomological dimension over $R$") to be the maximum of the integers $m$ such that $H^m(X; R)\neq 0$, where $H^*$ denotes the Alexander-Spanier cohomology. Now ...
0
votes
0answers
13 views

Non homeomorphic spaces with same homology groups [duplicate]

Is it possible for two spaces X and Y to have the same homology groups with X not homeomorhpic to Y.
-1
votes
0answers
30 views

semicontinious functions in topology space [closed]

Problem is: Let $X$ be a topologicyl space.. Prove that: A function $f\colon X\to\mathbb R$ is continuous function if and only if $f$ is lower semi-continuous and upper semi-continuous. The ...
2
votes
1answer
79 views

Do finite products commute with colimits in the category of spaces?

Let $X$ be a topological space. The endofunctor $\_\times X$ of the category of all topological spaces does in general not possess a right adjoint, since the category is not cartesian closed. Is it ...
2
votes
0answers
31 views

Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq ...
5
votes
1answer
53 views

Bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_n/\pi_1$

I would like to prove that there is a bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_{n}(X,x_{0}) / \pi_{1}(X,x_{0})$ where the action of $\pi_{1}(X,x_{0})$ over ...
1
vote
0answers
15 views

Prove a monomial is admissible

Set $\mathcal{F}_k\subset P_{k-1}$, where $$\mathcal{F}_k = \{x_j^2x_{j_1}x_{j_2}\ldots x_{j_{k-3}}: 1\leqslant j_1 < j_2 < \ldots < j_{k-3}<k, \ 1 \leqslant j <k\}$$ Show that, let ...
2
votes
0answers
18 views

Is a finite cyclic group a Poincare duality group?

I am trying to understand whether the finite cyclic group of order $n$, $C_n$ is a Poincare duality group, i.e. whether it's classifying space $K(C_n,\,1)$ is a Poincare complex. I know that the ...
0
votes
1answer
44 views

relative homotopy groups

I study relative homotopy groups and I have a question: Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which ...
2
votes
1answer
64 views

Section 22 in Munkres' TOPOLOGY, 2nd edition: How to establish this equivalence?

Let $X$ and $Y$ be topological spaces; let $p \colon X \to Y$ be a surjective map. Then $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is ...
3
votes
2answers
46 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way? [duplicate]

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
2
votes
0answers
66 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
2
votes
1answer
58 views

Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
4
votes
1answer
28 views

Completing compact surfaces with boundary to closed surfaces in $\mathbb R^3$

My question is whether any compact smooth surface in $\mathbb R^3$ (with smooth boundary) can be completed to a closed smooth surface in $\mathbb R^3$ without boundary? It is easy to complete it to an ...
3
votes
1answer
21 views

how to compute the de Rham cohomology with compact support of a mobius strip

I am having problem computing the de Rham cohomology with compact support of an open mobius strip,it's aquestion from Bott's book, and Bott said its cohomology is identically zero which can be ...
5
votes
2answers
78 views

Finding the de Rham cohomology of an open subset of $ \Bbb{R}^{n} $ minus a point.

Here’s my question: Let $ n \in \mathbb{N}_{\geq 2} $. Suppose that $ U \subseteq \Bbb{R}^{n} $ is an open set and that $ x \in U $. Then show that $$ {H_{\text{dR}}^{n - 1}}(U \setminus \{ x ...
2
votes
1answer
32 views

Join of closed embeddings is a closed embedding

An exercise from James's book General Topology and Homotopy Theory asks the reader to prove that if $\phi_1:X_1 \to Y_1$ and $\phi_2:X_2 \to Y_2$ are closed topological embeddings, then $\phi_1 * ...
0
votes
0answers
54 views

Cohomology ring of $S^3 \setminus A $ and $S^3 \setminus B $,where $A$ is union of two once linked circle and $B$ is union of two unlinked circles

Suppose $A$ is union of two once linked circles in $S^3 $ and $B $ is union of two unlinked circles.show that $S^3 \setminus A $ and $S^3 \setminus B$ have same cohomology group but not same ...
-3
votes
0answers
60 views

Cup product Structure of $X \vee Y $

Suppose $\alpha \in H^*(X)$ and $\beta \in H^*(Y)$ are of positive degrees. Show that $\alpha\beta=0$ in $H^*(X \vee Y)$. I am unable to show that. I think $\alpha\beta=0 $ because intersection of ...
1
vote
1answer
42 views

What exactly is the Kahler class of a torus?

Is there an intuitional way to understand what the Kahler class of $T^2$ actually is? It would be extremely useful to me if you could provide me some intuition behind it!
2
votes
2answers
38 views

Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
1
vote
1answer
48 views

computing Lefschetz number

We have a fixed point theorem which says that : Let $X$ be a compact polyhedron, $f:X\rightarrow X$ be a continuous map. If $L(f)\neq 0$ then $f$ must have a fixed point. (Lefschetz number is ...