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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3
votes
2answers
41 views

Non-orientable manifolds and mod 2 homology

I was reading the wonderful book "The wild world of 4-manifolds" by Alexandru Scorpan and I found the following sentence: "We are able to orient $\mathfrak{M}$ (else we only get modulo 2 ...
0
votes
0answers
11 views

extending map from quotient spaces to pairs

Suppose $(X,A)$, $(Y,B)$ are compact pairs of $\mathbb{R}^n$ and $f: (X/A,[A]) \to (Y/B,[B])$ is continous. Is it possible to find a map $\hat{f}:(X,A) \to (Y,B)$ such that $f([x]) = [\hat f(x)]$?
1
vote
1answer
25 views

What are higher dimension analogues of loops called?

A path $f:I\to X$ with the same starting and ending point $f(0)=f(1)=x_0\in X$ is called a loop. What is the higher dimensional analogue of a loop $f: I^n\to X$ called?
0
votes
0answers
23 views

what's the universal covering group of general linear group $GL(n,\mathbb{R})$ and $GL(n,\mathbb{C})$

Because the general linear group is not simply connected, they must be able to be covered by some simply connected Lie group. But I cannot imagine which Lie group with same dimension can cover ...
1
vote
1answer
46 views

What is $\beta_h$ in Hatcher?

I am puzzled what does $\beta_h$ refer to in this paragraph in Hatcher's book Algebraic Topology. Any idea? Is it the induced homomorphism of $h$? Ok, I found it in page 28. $\beta_h$ is the ...
1
vote
0answers
15 views

Working with Triangulations

I would like to prove that $\chi _(M_1\# M_2)=\chi(M_1)+\chi(M_2)-2$. However, the notes I'm using only mention following statement: Let $T_1,T_2$ be two finite triangulations of a compact ...
3
votes
0answers
34 views

Is $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$?

I was reading a little about how to imagine the projective plane and I have some weird intuition that says $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$. Is this true, and if ...
0
votes
0answers
16 views

Cohomology of Grassmanian: pairing with fundamental class

Let $Gr(k, V)$ be a Grassmannian with $\dim V=n$, and $S$ be a tautological bundle over $Gr(k, V)$, so $\operatorname{rank} S=k$. Then the cohomology ring $H^*(Gr(k, V))$ is generated by Chern classes ...
2
votes
2answers
51 views

Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
0
votes
0answers
31 views

Homeomorphic compact spaces have homeomorphic boundaries

Let $X$ and $Y$ be open bounded subsets of $\mathbb{R}^n$. How to show that $\bar{X}$ and $\bar{Y}$ are not homeomorphic whenever $\delta X$ is not homeomorphic to $\delta Y$? I know that if $\delta ...
4
votes
0answers
78 views

Different Proof for $\mathbb{R}^m\cong \mathbb{R}^n$ if and only if $n=m$.

Using homotopy it is easy to prove that (in topology) $\mathbb{R}^n\cong \mathbb{R}^m$ if and only if $n=m$. This result seems intuitively true, but, as realized very earlier and almost everyone who ...
1
vote
1answer
43 views

Homotopy groups of compact surfaces

I want to calculate the higher homotopy groups of $\Sigma_g$ and $\mathbb{R}P^2\# \mathbb{R}P^2\#\cdots\# \mathbb{R}P^2$. But I haven't found the methods to calculate the homotopy groups of connected ...
4
votes
1answer
55 views

The classifying space of open covers of a manifold

Let $M$ be a manifold of dimension $d$ and let $\mathsf{Disk}_{/M}$ be the category of open subsets of $M$ that are diffeomorphic to $\mathbb{R}^d$ with morphisms given by inclusions. Let $\mathrm{B} ...
2
votes
1answer
27 views

Infinite wedge sum of circles and a space homotopy equivalent

Given a space $G = \bigcup_{n=1}^{\infty} A_n$ where $ A_n $ is a circle $ C[ (n,0),n] \in R^{2}$ I'd like to show that its fundamental group is an infinately generated free group. So let's say $a_i $ ...
0
votes
0answers
32 views

Fundamental group and universal cover for this quotient space

For a non-zero integer $p$ define the topological space $L_p$ by: \begin{equation} L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p} \end{equation} Check that $L_1\cong\mathbb{D}^2$, and more ...
0
votes
0answers
17 views

Why is the homomorphism $\Omega_* \to H_*(BSO, Q)$ well defined.

Let $\Omega_*$ be the oriented bordism ring of a point. The correspondence of the map in the title is defined by $(K_{M^{N-1}})_*[M^{n-1}]$ where $K_{M^{N-1}}$ is the classifying map of the tangent ...
0
votes
0answers
13 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
votes
1answer
32 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 ...
0
votes
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
votes
1answer
54 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 ...
1
vote
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 ...
0
votes
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
votes
0answers
35 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 ...
0
votes
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
votes
2answers
45 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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
3answers
66 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 ...
1
vote
3answers
36 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
votes
2answers
156 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, ...
0
votes
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 ...
0
votes
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 ...
1
vote
0answers
30 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 ...
1
vote
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
votes
1answer
38 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 ...
0
votes
2answers
29 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
1answer
23 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 ...
0
votes
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 ...
1
vote
0answers
22 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
vote
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 ...
0
votes
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
vote
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
votes
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
votes
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 ...
0
votes
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 ...