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

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15 views

Construction of the dunce hat and homotopy between maps

The dunce cap results from a triangle with edge word $aaa^{-1}$. I was reading this answer and it is said that: We have that the dunce cap is constructed by gluing $D^2$ to $S^1$ under the map $g: ...
2
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1answer
35 views

Representing the $2$-homology classes of a $4$ manifold. Last passage of a Proof

I found a few occurrences of the same proof about representability of elements of $H_2(M,\mathbb{Z})$ for $M$ a closed orientable smooth $4$-manifold. All of them stop at the very end claiming that ...
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0answers
19 views

Persistent Homology. Missing points

I'm working on a project with a professor. This project involves Persistent Homology methods over a point cloud. Recently we found some inconsistencies in the point clouds that we were reading, ...
2
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1answer
55 views

Image of divisors under étale cover

Let $f\colon X\to Y$ be an étale cover of degree $d$ between two smooth projective varieties. If $V\subset X$ is an effective reduced and irreducible divisor, does $f$ restrict to an isomorphism ...
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1answer
26 views

Examples in which contractibility is not closure-preserved, not interior-preserved, not intersection-closed or not connected union-closed.

I read the following 4 theorems: http://topospaces.subwiki.org/wiki/Contractible_space (1) It is possible to have a topological space $X$ and a subset $A$ of $X$ such that $A$ is contractible in the ...
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1answer
21 views

What is the best way to understand the uniform topology generated uniform metric?

In $\Bbb{R}^\infty$ ($\Bbb{R}\times \Bbb{R} \times \Bbb{R} \times \Bbb{R} \times ....$), What is the basis element generated by unifrom metric with radius 1? I think (-1,1) x ( -1,1) x (-1,1) .... ...
2
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0answers
36 views

stable homotopy groups of the projective plane

The projective plane $\mathbb{R}P^2$ is obtained by attaching a $2$-cell to $S^1$ via a degree $2$ map $$ S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2. $$ Question 1: the ...
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0answers
18 views

Contentedness of subspace Y of X in general case.

If Y is a subspace of X, a separation of Y is a pair of disjoint non-empty sets A and B whose union is Y. From above statement, I am not sure whether pair of disjoint non-empty sets A, B are open ...
2
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2answers
46 views

Why are bordism groups of a point nontrivial

My definitions of Bordism are from Tom Dieck's Algebraic topology book. Briefly, a singular manifold, $M \xrightarrow{f} pt $, for a closed smooth oriented manifold $(M, \omega)$ without boundary is ...
4
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0answers
25 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces ...
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1answer
61 views

if M is compact and N is connected, then M=N …?

Let M and N be surfaces in $R^3$ such that M is contained in N. If M is compact and N is connected, prove that M=N. ================================= I thought intuitively the compactness means ...
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0answers
46 views

Free homotopies and extensions

I am trying to prove the following. Lemma. Let $X^n$ be the $n$-skeleton of a CW complex $X$ with attaching functions $\phi_{\beta}:S^{n-1}_{\beta} \to X^{n-1}$, for all $\beta \in B$, and let ...
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0answers
20 views

Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic ...
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3answers
67 views

Is $\mathbb{R^2}$ Hausdorff? Give an example of a non-Hausdorff topology on $\mathbb{R}$

these are two questions on Hausdorff topological spaces. The bit I am having particular difficulty with is finding an 'example of a non-Hausdorff topology on $\mathbb{R}$' A Hausdorff topological ...
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3answers
37 views

Are these metrics?

I want to find if the below functions are metrics. I have worked through each of the three conditions, but am stuck on the positivity of $f(a, b)$ (first condition-see below) and the triangle ...
11
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2answers
157 views

Why are we interested in cohomology?

I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, ...
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0answers
43 views

Homomorphisms of Chain Complexes

Let $(K, d^{K})$ and $(L, d^{L})$ be chain complexes. For $n \in \mathbb{Z}$ define $$ \mathrm{Hom}(K, L)_{n} := \prod_{j \in \mathbb{Z}} \mathrm{Hom}(K_{j}, L_{j+n})$$ and $$ d_{n}^{K,L} \ \colon ...
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0answers
9 views

Understanding uniform metric and uniform topology

$$ p(x,y) := \sup_{\alpha\in J} \min(1, |x_{\alpha}-y_{\alpha}|) $$ for any two points $x := (x_{\alpha})_{\alpha\in J}$, $\,$ $y := (y_{\alpha})_{\alpha\in J}$ $\,$ in $\mathbf{R}^J$. Above is the ...
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0answers
31 views

Why must the vertices of a delta complex be ordered?

We can compute the homology of nice spaces by putting either a delta complex or simplicial structure on that space and computing the homology of the associated chain complex. With a simplicial ...
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1answer
26 views

An odd map having odd degree little issue with the proof

In the proof below of hatcher I agree that $Im(\tau) \subset Ker(p_{\#})$. However, what I don't understand why is the other inclusion true ?
2
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1answer
39 views

Is the Klein bottle homeomorphic to the union of two Mobius bands attached along boundary circle?

Question: Determine whether the Klein bottle is homeomorphic to the union of two Mobius bands attached along their boundary circles. The Klein bottle is the quotient space $$ K=I^2 /{\sim}, \quad ...
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2answers
30 views

Covering spaces of wedge sum of circles

Let's suppose I'd like to characterise all connected covering spaces of wedge sum of circles.When it comes to a torus, a sphere or $ S^{1} \vee S^{2} $ it is easy to point out exactly how do all ...
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1answer
27 views

Homotopy classes of manifolds of dimension $2n$ which are $(n-1)$ connected

In his essay "Classification of (n − 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres" Milnor states (in passing) "The manifold can be built up (up to homotopy type) by taking ...
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1answer
31 views

Locally compact Hausdorff space is metrizable

Given $X$ a Hausdorff space, I have a hunch that $X$ is locally compact $\iff X$ is metrizable. I am not sure if it is true because I do not know how to prove that. To prove the implication ...
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1answer
23 views

Example that the topology of $|K|$ is larger than the topology $|K|$ inherits as a subspace of $\mathbb{R}^n$

I'm a lil bit confused with the example 3 from Munkres, that the topology of $|K|$ is larger than the topology $|K|$ inherits as a subspace of $\mathbb{R}^n$. Let $K$ be the collection of 1-simplices ...
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1answer
24 views

Homeomorphism of $SO(3)$?

I am trying to get a better understanding of the homeomorphism of $SO(3)$ to the Real Projective Plane, so that ultimately I can show that $\pi_1(SO(3)) = \mathbb{Z}_2$. From wikipedia and many other ...
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0answers
22 views

Difference between simplicial complex and underlying space

An underlying space $|K|$ is the subset of $R^n$ which is the union of the simplices of $K$. While a simplicial complex $K$ is a collection of simplices s.t. every face of its simplex is in $K$ and ...
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0answers
23 views

Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
1
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1answer
24 views

What is the map from $H_j( \Sigma MSO(k)) \to H_{j-k}(BSO(k))$ on Tom Dieck page 537

I am reading Tom Dieck's page 537 and I am not sure what the vertical map that I put in the title is in the diagram in the bottom of the page. This map is labeled Thom Isomorphism. Here $MSO(k)$ is ...
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0answers
46 views

degree of $f\circ g$

Let $f,g : \mathbb{S}^1 \to \mathbb{S}^1$ be two continuous maps where $\mathbb{S}^1$ is the unit circle. Prove that $\deg (f \circ g) = \deg f \deg g$. I don't know at all how to do it : I first set ...
1
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1answer
31 views

Abelianization of free product is the direct sum of abelianizations

I define $\text{Ab}(G)=G/[G,G]$ where $[G,G]$ is the commutator subgroup. I want to show that $$\text{Ab}(G_1*G_2)\cong \text{Ab}(G_1)\oplus\text{Ab}(G_2)$$ This page gives a categorical proof, but I ...
2
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1answer
46 views

Find the rank and the free generators

Consider the homomorphism$ \ $ $f:\ F\{x,y\} \to <x,y|x^2, y^3, xyx^{-1}=y^{-1}>$, find the free generators of $kerf$. I know that we should first consider the wedge sum of circles whose ...
2
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1answer
30 views

Clarification on Relative homology

Suppose we have topological spaces $A$, $B$ $\subset$ X. Further assume $A$, $B$ are homeomorphic. Then shouldn't it directly follow from the definitions that $H_{n}(X,A)$ and $H_{n}(X,B)$ are ...
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0answers
13 views

Construction of Moore-Postnikov Tower

Given a map $f:X\to Y$, a Moore-Postnokiv tower is a sequence of spaces $...\to Z_3 \to Z_2 \to Z_1$ together with maps $X \to Z_n$ inducing isomorphism on $\pi_i$ for $i<n$ and surjection for ...
5
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1answer
76 views

Existence of incompressible surface in a non-orientable manifold.

Let $M$ be a compact $P^2$ -irreducible 3-manifold. If $M$ is non-orientable, then there is a compact surface $F$ properly embedded in $M$ such that $F$ is two-sided, non-separating and ...
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0answers
20 views

Intersections of a ray from star domain to its boundary

Let $S \subset \mathbb{R}^n$ be bounded, open, star domain (or star-shaped) relative to $x_0$. A ray $R$ from $x_0$ is the set of all points $x_0+tp, p\in\mathbb{R}^n-0, t\in\mathbb{R}_{\geq0}$. It ...
5
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2answers
233 views

Space which is connected but not path-connected

Consider the following two definitions: Connected : A topologiocal space X is connected if it is not the disjoint union of two open subsets, i.e. if X is a disjoint union of two open sets A and B, ...
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0answers
28 views

Remark in Hatcher's Algebraic Topology on Mapping Cylinders

On p2 of Algebraic Topology, Hatcher defines mapping cylinders as follows: For a map $f: X \to\ Y$, the mapping cylinder $M_f$ is the quotient space of the disjoint union $(X \times I) \cup Y$ ...
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1answer
27 views

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$?

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$? $\pi(X)$ denotes the fundamental group of $X$.
3
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1answer
64 views

Question about n-simplex and its face

I just started to read Elements of Algebraic Topology by Munkres and got question already. On page 5, (Let $\sigma$ be an n-simplex) since Bd $\sigma$ consists of all points $x\in\sigma$ s.t. at least ...
2
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2answers
150 views

existence of a lifting

I was asking myself if given a covering projection $p: X' \rightarrow X$ and a continuous map $f:Y \rightarrow X$, does a lift $f: Y \rightarrow X'$ always exists? If no, could you please exhibit a ...
3
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1answer
60 views

Does trivial cohomology imply trivial homology? Does $\operatorname{Hom}(A,\mathbb Z) = \operatorname{Ext}^1(A, \mathbb Z) = 0$ imply $A = 0$?

Is there a topological space $X$ such that $H^i(X; \mathbb{Z}) = 0$ for all $i > 0$, but $H_n(X; \mathbb{Z}) \neq 0$ for some $n > 0$? In his answer to the question Is homology determined ...
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1answer
79 views

I cannot understand an explanation why 2-sphere is simply connected.

I am studying Elementary Differential Geometry written by Barrett O'Neill. In page 188, Chapter 4.7, there is an explanation why 2-sphere is simply connected. The following is from the text : ...
1
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1answer
41 views

Obstructions to putting a complex structure on a real vector bundle (other than, obviously, dimension)

A complex vector bundle is usually described as one with structure group $GL(n,\mathbb{C})$. If I take a real $2n$ bundle is it always the underlying real bundle of some complex bundle?
2
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1answer
31 views

figure-8 knot complement

The figure-8 knot seen as a 2-bridge knot with two maxima and two minima of the height function, has a complement in $S^3$ with one 0-handle,two 1-handles, two 2-handles and a 3-handle which cancels ...
1
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1answer
26 views

The induced homomorphism on the cohomology under the pinching map $S^n \rightarrow S^n \vee S^n$

There is a continuous map from $S^n$ to the wedge sum of spheres. I imagine that the induced pull back map on cohomology groups $H^n( S^n \vee S^n) \cong \mathbb{Z} \oplus \mathbb{Z} \rightarrow H^n( ...
0
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1answer
15 views

Why is $BSO(n-1)$ the sphere bundle of the tautolocigal bundle on $BSO(n)$?

To try to show this I wrote down explicitly what the classifying spaces can be realized as. I am realizing the classifying spaces $BSO(n-1)$ as $V^\infty_n \times_{SO(n-1)} pt$, where ...
7
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0answers
116 views

cohomology ring of a configuration space

Consider the following configuration space of triples of points. $C = \left\lbrace (z_1,z_2,z_3) \in (\mathbb C^*)^3, z_1 \ne z_2, z_1 \ne z_3, z_2 \ne z_3\right\rbrace - \left\lbrace (z_1,z_2,z_3) ...
0
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1answer
29 views

Why is homology $0$ here? $H(\vartriangle,A) = 0$

In the remark, it says the homology is $0$. I deduced that $$H_i(\vartriangle,A) = \frac{\ker(d_{-1})}{\operatorname{im}(d_0)} = \ker(d_{-1})$$ because $\operatorname{im}(d_0) = 0$. But why is ...
-1
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1answer
47 views

Computing Fundamental group [closed]

$a,b,c\in S^2$ and $X$ is a quotient space that make these three points as the same. Computing its fundamental group.