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

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1answer
49 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
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1answer
46 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
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2answers
54 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 ...
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1answer
24 views

Problem with Massey's exercise 3.3 [closed]

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 ...
7
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1answer
58 views

Boundary of boundary of singular cube is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
1
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1answer
65 views

Covering $S_2$ with $S_3$(or $S_n$)

How can I construct a covering map $p : S_3 \to S_2$? I can construct coverings for $S^1\vee S^1$, but same ideas don't work for $S_2$ ($S_g$ is sphere with $g$ handles).
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1answer
44 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
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1answer
40 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
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0answers
62 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
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0answers
74 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 $A \to X$ ...
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0answers
21 views

Long exact sequence of homotopy groups $\pi_n$ for a pointed homotopy pullback square

Let \begin{align} A &\to B\\ \downarrow &~~~~~\downarrow\\ C &\to D \end{align} be a homotopy pullback square of pointed simplicial sets. One gets a long exact sequence $$ ...
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0answers
50 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
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1answer
51 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
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1answer
50 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
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1answer
37 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
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2answers
103 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
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2answers
50 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
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2answers
64 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
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0answers
56 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
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1answer
52 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 ...
3
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1answer
59 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
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1answer
68 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 ...
2
votes
1answer
54 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
64 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
35 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
285 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 ...
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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 ...
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0answers
15 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.
3
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1answer
85 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
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0answers
33 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
59 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 ...
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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 ...
3
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1answer
27 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
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1answer
48 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
66 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
53 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
69 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
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1answer
62 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
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1answer
29 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
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1answer
27 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
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2answers
84 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 * ...
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0answers
61 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 ...
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0answers
61 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
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1answer
45 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
41 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
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1answer
51 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 ...
4
votes
1answer
108 views

If $f:S^1\to S^1$ doesn't have any fixed point then it is homotopic to the identity

How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
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0answers
36 views

thorough proof of Mayer-Vietoris implies Excision

Where could I find a very complete proof of how the Mayer-Vietoris sequence implies the Excision theorem? I've read a few proofs, but they always leave out the details! Thank you!