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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31 views

Understanding the cup product definition

I am trying to learn a little cohomology, and am having some trouble with this definition of the cup product that I found: $$(f\cup g) (\sigma) =f\left(\sigma_{[v_0, \ldots, v_k]} ...
3
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1answer
35 views

H-space multiplication question (homotopy)

Let $(X,x_0)$ be a $H$-space with multiplication $\mu:X\times X\to X$. Let $e$ denote the constant map $I^n\to x_0$. Is it true (and why) that $\begin{cases} ...
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0answers
22 views

Getting topological objects from the “cube” of $T^3$

One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$. To get ...
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32 views

two non-diffeomorphic manifolds with the same cohomoly classes.

Question: Calculate the de Rham cohomology groups of: $U=\mathbb{R}^3 - (L \cup C)$ and $V= \mathbb{R}^3 - (L' \cup C)$, where $L' = \{x = 2, y = 0\}$, $L = \{x = y = 0\}$ and $C=\{ x^2 + y^2 = 1, ...
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1answer
25 views

Homology of contractible space

I understand that if $f,g: X \to Y$ are maps and $f$ is homotopic to $g$, then the induced maps on the homology groups $f_*$ and $g_*$ are equal. Why does this imply that if $X$ is contractible then ...
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1answer
20 views

Two questions about a CW-complex example

Consider the CW-complex structure of X: Delete the interiors of two disjoint subdiscs in the interior of $D^2$, and then identify all three resulting boundary circles via homeomorphisms preserving ...
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21 views

Show that $ \operatorname{Sp}(1)=\{A \in M_1(\mathbb{H})\mid AA^*=I=A^*A\}=S^3 $

Let $M_n(\mathbb{H})$ be the set of all $n \times n$ matrices with entries in the quaternions $\mathbb{H}$. For $A=(a_{ij} ) $ let $ A^*=(a^*_{ij} ) $ be the matrix with $a^*_{ij}=\bar{a}_{ij} $, ...
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93 views

why covering dim of $n$-dimensional simplicial complex is at most $n$?

I'm self-studying "Elements of Algebraic Topology" by Munkres these days. In that book, Exercise #2 (d) in section 15 asks you to prove the following If $K$ is $n$-dimensional finite simplicial ...
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10 views

Homeomorhism of pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$

I'm having trouble following an argument in Bredon's Topology and Geometry. He says that the pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times ...
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5 views

Bottleneck Distance Significance?

Let $X$ be a smooth manifold and $f,g:X\rightarrow \mathbb{R}$ two real valued functions on $X$. Suppose we have two persistence diagrams $Dgm(f)$ and $Dgm(g)$ encoding the lifetime of $k$-dimensional ...
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1answer
15 views

Condition on compact subsets implying null singular homology

This is a review question I'm doing for an upcoming exam. Consider $X$ a Hausdorff space, and $(H_*(X),\partial_*)$ its singular homology. I must prove that, given $[z]\in H_p(X)$ there exists a ...
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0answers
12 views

Properties of free suspensions and free cones

In the category of based topological spaces, suspension $- \wedge S^1$ is adjoint to loop spaces $\text{Hom}(S^1,-)$ and the based cone $- \wedge I$ (where $I$ has zero as basepoint) is adjoint to the ...
2
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1answer
28 views

simplicial complex $K$ whose underlying space is homeomorphic to the cylinder (Example 1 page 16 Munkres)

I would like to ask about the Example 1 of Munkres' Elements of Algebraic Topology: Suppose we wish to indicate a simplicial complex $K$ whose underlying space is homeomorphic to the cylinder. Two ...
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0answers
16 views

Why is are the simplicial 1-chains $[A,B] \neq -[B,A]$?

This is a really simple question that I think I have answered, but I'm not altogether satisfied and would like confirmation or an alternative. We define the simplices $$ \begin{align} [A,B] : ...
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0answers
22 views

Show that $ \operatorname{Sp}(n)=\{A \in M_n(\mathbb{H}) \mid AA^*=I=A^*A\} $ is a compact group

Let $M_n(\mathbb{H})$ be the set of all $n \times n$ matrices with entries in the quaternions $\mathbb{H}$. For $A=(a_{ij} ) $ let $ A^*=(a^*_{ij} ) $ be the matrix with $a^*_{ij}=\bar{a}_{ij} $, ...
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0answers
36 views

Prove there is a point $y$ such that $ g(y)=0 $ where $ g: \bar{D}^2 \rightarrow \mathbb{C} $

Let $ g: \bar{D}^2 \rightarrow \mathbb{C} $ be a continuous function on a closed disk satisfying $ g(-x)=-g(x) $ for any $ x \in \partial \bar{D}^2$ Prove that there is a point $ y \in ...
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1answer
40 views

Quotient map that is not closed

Can anyone help me find some example of a closed relation $\sim$ on a Hausdorff space $X$ such that the quotient map $p:X→X/\sim$ is not a closed map? Here an equivalence relation $\sim$ is closed if ...
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1answer
33 views

Question on the proof that underlying space $|K|$ is Hausdorff in Lemma 2.4 from Munkres

In Lemma 2.4 (Munkres), showing that underlying space $|K|$ is Hausdorff (given a simplicial complex $K$), if $x_0\neq x_1$ there's at least one vertex $v$ s.t. $t_v(x_0)\neq t_v(x_1)$. Can anyone ...
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14 views

What is bottleneck distance intuitively?

Can someone explain the intuition behind Bottlneck and Wasserstein distance? The context here is the comparison of two persistence diagrams.
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30 views

computing $\pi_1 S^1$ from a spectral sequence

In most of my calculations that I have done based off of mosher and tangora, calculations have proceeded by knowing for example that the fiber of some fibration is say a $1-sphere$. From this we are ...
2
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1answer
31 views

Orientation on the boundary of a manifold

Let $M$ be a manifold with boundary. Hatcher writes that a compact manifold with boundary is $R$-orientable if $M - \partial M$ is $R$-orientable. That is there exists a function $x \to \mu_x \in ...
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2answers
61 views

Two lifts of a local homeomorphism

Just learning about sheaves. Suppose I have a sheaf $\mathscr{F}$ on a topological space. (The sheaf can take values in sets, let's say.) Let $E \overset{\pi}{\to}X$ be the etale space of this ...
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14 views

Compact Vertical Cohomology and Euler Class of CP1

First of all, please excuse my English. I'm not native Englsih speaker, so you will see so many grammer mistakes, I just only hope that my mistkae wouldn't effect what I want to mean. Hi, recently ...
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0answers
17 views

Group Operation in $\pi_n$ defined using H-space

This is a question from Hatcher: For an $H$-space $(X,x_0)$ with multiplication $\mu:X\times X\to X$, show that the group operation in $\pi_n(X,x_0)$ can also be defined by the rule ...
2
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1answer
36 views

Is an underlying space of a simplicial complex second countable

Suppose given a simplicial complex $X$. The underlying space $|X|$ is the subset of $\mathbb{R}^n$ which is the union of simplices of $X$. Is $|X|$ is second countable? I know that any Euclidean ...
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2answers
20 views

Euler characteristic of connected sums of projective planes and tori?

For homework, I was told to prove the following equalities. $$\begin{gather} \chi _{i=1}^n(\# T^2 _i)=2-2n \\ \chi _{i=1}^n(\# \mathbb RP^2 )=2-2n \end{gather}$$ First of all, the notation is strange, ...
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2answers
92 views

The Modules over Algebras over Operads are not what they seem.

Operads are a nice framework to model all kinds of different algebras, i.e. Monoids are algebras over the operad Assoc in the category of sets Associative algebras are algebras over the operad Assoc ...
2
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1answer
30 views

Prove that $W \cup S^1$ is connected in the subspace topology of $\mathbb{R^2}$

I want to solve the following question: Prove that the union of $W$ and the unit circle $S^1$ is connected in the subspace topology of $\mathbb{R^2}$ where $W=\{(x, y) \in \mathbb{R^2} | ...
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0answers
14 views

Is this covering group $G'$ of $G$ unique?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
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1answer
32 views

configuration spaces in mathematics and in physics

On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics. Question. Do ...
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1answer
26 views

$d(a,X) = d(a,\overline{X})$ (distance from point to set is distance from point to closure)

I'm trying to understand this proof that: $$d(a,X) = d(a,\overline{X})$$ The proof says: Since $X\subset \overline{X}$, then $d(a,\overline{X})\leq d(a,X)$. We just need to show that the $<$ ...
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1answer
20 views

Is the fundamental group of a retract a subgroup of the original space?

Let $X$ be a topological space and $A$ a retract of $X$. Is the fundamental group of $A$ a subgroup of the fundamental group of $X$?
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1answer
81 views

Why is $\mathbb{Z}$ path-connected?

I understand a space $X$ is path-connected if there exists a path $\tau$ for every point $x_1,x_2 \in X$ such that $\tau(0)=x_1,\tau(1)=x_2$. And a path must be continuous. Well, I cannot find a path ...
2
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1answer
102 views

Badly explained solution

My algebraic topology class is very bad at teaching, it just doesn't explain what's needed. Let me be specific, I am looking at this question, Q. Find the degree of $f_0 :S^1 \to S^1$ the constant ...
2
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0answers
42 views

Topological book which covers applications in the Medical Field (Medicine/Bacteria/Cancer/Virues)

To get to the point I'm looking for a book on Topology that covers specifically it's uses in the medical field. I've seen a lot of book requests in Topology, but they are all about learning topology ...
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33 views

Does $\#_n S^2 \times S^1$ really admit a map of non-zero degree from $B \times S^1$

In this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ (the connected sum of two ...
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2answers
108 views

Betti numbers of complex “sphere”

Let $X$ be the set of solutions to $x_1^2+\ldots+x_n^2=1$ in $\mathbb{C}^n$. This has real dimension $2(n-1)$, but since $X$ is an affine algebraic variety, the only possible non-zero topological ...
3
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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 ...
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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 $f([x]) = [\hat f(x)]$?
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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?
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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
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0answers
76 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 ...
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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
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1answer
53 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} ...