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

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Gromov's defenition of Content of Ball

Let $B(p, R)$ denote the metric ball of radius $R$ centered at $p$ in a manifold. Then Gromov defined the content of the ball by $$Cont(B(p,R))=rank(H_*(B(p, R/5))\to H_*(B(p,R))) $$ and he remark ...
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1answer
20 views

Intersection of an open neigbourhood of a mobius strip and and a disk

I have seen that the real projective plane of dimension 2 can be expressed as a union of a disk and a mobius strip but what I don't understand is that why their intersection is a cylinder? Can ...
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1answer
31 views

Stiefel-Whitney Numbers of $\mathbb{R}P^2\times \mathbb{R}P^2$

I'd like to calculate the Stiefel-Whitney numbers of $\mathbb{R}P^2\times\mathbb{R}P^2,$ but don't know how to. My first instinct was to say that the tangent bundle is isomorphic to the product of ...
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1answer
70 views

Prove or disprove any continous map $f$ from $T^2$ to $RP^2$ is null-homotopic.

I tried to solve the following question: Prove or disprove any continous map f from $T^2$ to $RP^2$ is null-homotopic. We know the universal cover of $RP^2$ is $S^2.$ I want to construct a map $g$ ...
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1answer
33 views

Direct proof of decomposition of real vector bundle of odd degree into the direct sum of a trivial bundle and another of even degree

The real splitting principle tells us that when taking a real, oriented vector bundle of odd dimension $\zeta$ over a manifold $M$ you can always write $\zeta$ as $\tilde{\zeta} \oplus \varepsilon^1$, ...
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1answer
55 views

Is homology determined by cohomology?

I am aware of the universal coefficients theorem for cohomology which implies that the homology groups completely determine the cohomology groups. I am wondering if cohomology determines homology in ...
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1answer
28 views

Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$.

Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$. The problem is in page 457 of "Topology and Geometry" written by Glen E. Bredon. I think I need to use a long exact sequence of ...
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1answer
42 views

Fundamental group of a quotient of $S^2 \times I$.

Let $X=S^2 \times [0,1]$ and let Y be the quotient space obtained from X identifying each point $x\in S^2 \times \{1\}$ with its antipodal in $S^2 \times \{0\}$. How can I calculate $\pi_1 (Y)$? All ...
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0answers
61 views

Fundamental group of projective plane with handles

I was told that the fundamental group of the projective plane with g handles is isomorphic to $\langle c_1, \ldots, c_{2g+1} | c_1^2 \cdot \ldots \cdot c_{2g+1}^2\rangle$. How can I show it? I can ...
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2answers
52 views

Fundamental group of two circles joined

If we consider the union of two $S^1$ joined at point, the fundamental group is a non abelian free group with two generators. Intuitively is clear but, how can I prove it formally? Thanks in advance. ...
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74 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 ...
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2answers
63 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 ...
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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 ...
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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? ...
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2answers
53 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
23 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 ...
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1answer
64 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 ...
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1answer
39 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 ...
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61 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 ...
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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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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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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 ...
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1answer
50 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 ...
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1answer
49 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 ...
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1answer
36 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 ...
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2answers
102 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 ...
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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 ...
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63 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 ...
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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 ...
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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 ...
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1answer
56 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, ...
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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 ...
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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 ...
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1answer
52 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 ...
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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 ...
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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 ...
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3answers
283 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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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.
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84 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 ...
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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 ...
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1answer
57 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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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 ...
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0answers
19 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 ...
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1answer
45 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 ...
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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 ...
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2answers
52 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} ...
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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 ...
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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}. ...