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
3answers
286 views

Knot Group and the Unknot

Hi I am stuck in trying to show that given a knot $K$ such that the knot group $\pi_1(K)=\mathbb Z$ then $K\simeq U$. I tried to use the fact that the infinite cyclic cover is the universal cover but ...
4
votes
2answers
88 views

What are polyhedrons?

Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ ...
4
votes
0answers
63 views

Generalization for Leray Hirsch theorem for Principal $G$-bundle

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
7
votes
0answers
154 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
0
votes
0answers
49 views

how to show whether this topological space is metrizable or not?

Let $X$ be a two-element topological space with a discrete topology. Let $J$ be an uncountable indexed set. And let $Z=X^J$ be the Cartesian product endowed with the product topology. Is $Z$ ...
1
vote
1answer
37 views

Two ways to split the second Betti number

The definition of the positive and negative parts of the second Betti number which I know is via the diagonalized intersection form, and possible for $4$-manifolds $M$. $b^\pm_2:= \dim H^2_\pm( M;\...
0
votes
1answer
28 views

Verification of a proof regarding the connected sum of two surfaces

I am trying to solve the following exercise: Let $X_1, X_2$ be two surfaces. Lets consider charts $\varphi_j: U_j \to \mathbb{R}^2$ with $U_j \subset X_j$, $j= 1, 2$ and let $B_j = \varphi^{-1}_j(...
2
votes
1answer
26 views

$\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus.

$\pi_1(\mathbb{T}^2\#\mathbb{T}^2) \cong <\beta_1, \gamma_1, \beta_2, \gamma_2|\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}\beta_2 \gamma_2 {\beta_2}^{-1}{\gamma_2}^{-1}=1>$ My question : ...
2
votes
0answers
19 views

Weak hausdorff but not compactly generated? [duplicate]

What is an example of a weak Hausdorff space that is not compactly generated? I can't think of any, and googling doesn't reveal anything...
10
votes
1answer
234 views

Self study Persistent Homology

I am a graduate student in mathematics interested in Persistent Homology. Can anyone recommend any good books / resources to self study Persistent Homology? I am taking a course in Algebraic ...
4
votes
0answers
109 views

Question about Gysin map (pushforward in cohomology)

The setting is the following: $X \subset \mathbb{CP}^r$ is an algebraic subvariety of dimension $n$, codimension $e=r-n$, and degree $d$. Call $j$ the inclusion. Then, Poincaré duality induces a map $...
2
votes
1answer
96 views

$G$-sets, natural correspondence?

See here. In the category of $G$-sets, the morphisms $f:G/H\to X$ are in one-to-one correspondence with the elements of $X^H$; the correspondence sends $f$ to $f(H)$ (where the subgroup $H$, being ...
6
votes
1answer
106 views

Consider $\mathsf{A,B,C},$ $\dots$ in a sans serif font. Each of these gives a graph in the plane. Sort these into homeomorphism classes.

Problem: Show that homeomorphism is an equivalence relation on topological spaces. Now consider the capital letters of the alphabet $\mathsf{A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z},$ in ...
5
votes
1answer
77 views

Showing that the Mapping Torus is a topological manifold

Let $X$ be a conected topological $n$-manifold, and $f:X\rightarrow X$ an homeomorphism, the Mapping Torus $M_f$ is defined as, $$M_f=X\times [0,1]/\sim$$ where $(x,0)\sim (f(x),1)$. I am trying to ...
1
vote
0answers
43 views

Behaviour of Chern class under deformation

The setting is the following: $f: X \rightarrow T$ is a smooth projective map of complex algebraic varieties, and $L$ is a line bundle on $X$. My question is the following: is $c_1(L_t)$ (in ...
3
votes
0answers
34 views

Equality $H^i(K,\mathcal{F}_{|K})=\varinjlim_{U\supset K}H^i(U,\mathcal{F}_{|U})$ for a constructible sheaf

The setting is the following. I have a complex algebraic variety $X$, and $\mathcal{F}$ is a constructible sheaf on it (i.e. there is a stratification of Zariski-locally closed subsets $X=\sqcup_{i \...
2
votes
1answer
42 views

Proof of : “Signature of $\mathbb{C}P^{2n}$ is $1$”

I started learning about signature of a $4k$-manifold and one of the most common example is the signature of $\mathbb{C}P^{2n}$. The only reference I found is tom Dieck's Algebraic Topology. Even ...
2
votes
0answers
30 views

Connect Sum of a connected, compact manifold of dimension n and $S^n$

$M $ be a connected,compact manifold of dimension n. Show that $ M \# S^n$ is homeomorphic to $M$ My idea: $S^n-D^n$ is homeomorphic to $D^n$..so $M \# S^n$ is homeomorphic to $(M-D^n) \cup D^n$ ......
0
votes
1answer
52 views

Triangle inequality of hyperbolic metric

For $z_1, z_2 \in \mathbb{B}^2$, define $d(z_1, z_2) = \text{cosh}^{-1}(1+ \dfrac{2|z_1 - z_2|^2}{(1-|z_1|^2)(1-|z_2|^2)})$. In my text book (Lee's Topological manifolds Problems 12-23), to prove ...
6
votes
2answers
111 views

Subspace of a compactly generated space?

Consider the following two exercises in May's A Concise Course in Algebraic Topology. (a) Any subspace of a weak Hausdorff space is weak Hausdorff. (b) Any closed subspace of a $k$-space is a $...
1
vote
0answers
42 views

What are the exact critera for a CW-complex being a polytope?

Everybody talks about the fact that polyhedra are special CW complexes, and some of the higher dimensional abstract polytopes are too, but nobody tells the exact criteria for a CW complex being a ...
2
votes
0answers
45 views

Is it true that Morse function on non-trivial knot has at least 4 critical points?

I'm actually interested in the continuous case, for a non-trivial knot $S^1\rightarrow \mathbb{R}^3 $ is it true that the function $\sin(t)$ can not extend to a continuous function on $\mathbb{R}^3 $?
0
votes
2answers
46 views

Rank of fundamental groups of open subsets.

Let $X$ be an open connected subset of the real plane. Then it is known that $\pi_1(X)$ is a free group. Is there a useful formula for the rank of $\pi_1(X)$? I suspect that the rank should be $b-1$, ...
5
votes
1answer
81 views

Euler Characteristic of a boundary of a Manifold

I need some guidance in understanding a specific passage of the following result taken from [tom Dieck Algebraic Topology, page 456] Proposition 18.6.2. Let $B$ be a compact $(n+1)$-manifold with ...
4
votes
2answers
108 views

Proper and discontinuous action of a group

I came across the following problem of algebraic topology that I couldn't solve. Let $\Gamma$ be a group which acts properly and discontinuously in a topological Haussdorf space $X$. Let $H \...
1
vote
2answers
130 views

Are all paths with the same endpoints homotopic in a simply connected region?

It is clear to me that if all paths (with the same endpoints) in a region are homotopic then that region is simply connected, however I am having difficultly proving the converse, that is, all paths ...
0
votes
0answers
44 views

Simpicial approximation map

Let $\Delta^n$ be the standard $n$-simplex and $f :\delta\Delta^n\to \delta\Delta^n $ ($\delta\Delta^n$ is a boundary of $\Delta^n$ ) be a continuous function such that $f(-x) = -f(x)$. Is there a ...
0
votes
0answers
23 views

Definition of General Position for a Semi-Algebraic Set

I'm looking for a precise definition of what it means for a semi-algebraic set to be in general position. I've found definitions that apply to sets of points, but that doesn't seem to help in the ...
1
vote
1answer
79 views

How can I draw (using a computer) spaces that I can't parametrize easily?

I am studying algebraic topology and I came around the following problem: I have to describe the space obtained when I identify the circles marked with different letters in the following figure: ...
1
vote
1answer
48 views

Homotopy equivalences between graphs realizing isomorphisms on $\pi_1$

my question is how to proof this statement: Any isomorphism $φ:π_1(G_1,u_1) \to π_1(G_2,u_2)$ can be realized by a homotopy equivalence $f:(G_1,u_1) \to (G_2,u_2)$ If $φ:π_1(G_1,u_1) \to π_1(...
0
votes
0answers
36 views

definition of finite pointed spaces in Lurie Higher Algebra

Let $S$ be the infinity category of spaces. In Higher Algebra 1.4.1.4 Lurie defines $S^{fin}$ as the smallest full subcategory of $S$ which contains the final object $*$ and is stable under finite ...
1
vote
2answers
47 views

Even dimensional real projective spaces cannot be combed

I have to prove that the even-dimensional real projective space cannot be combed, i.e. there isn't any non-vanishing smooth vector field. (I can't use Hopf theorem since those manifolds are not ...
1
vote
2answers
151 views

Practicing Seifert van Kampen

I am just practicing how to use Seifert-van Kampen. The following excercise is from Hatcher, p.53. Let $X$ be the quotient space of $S^2$ obtained by identifying the north and south poles to a ...
1
vote
0answers
23 views

unwinding the definition of $H_i(KU)$ as a map of spectra $\mathbb{S}^i \to HZ \wedge KU$

In the answer to this question on mathoverflow, it says The integral homology group $H_i(KU)$, the direct limit of $$\dots \to H_{2n+i}(BU)\to H_{2n+2+i}(BU)\to\dots,$$ My question is why? Here is ...
1
vote
1answer
52 views

Why is a Torus # Mobius strip $\cong$ Klein bottle # Mobius strip?

For simplicity, $T =$ torus, $M =$ Mobius strip, $K =$ Klein bottle, and $P = \Bbb RP^2$. I would like to know why $T \# M \cong K \# M$. I know that $K = P \# P$. The solution says $T \# M \cong T ...
1
vote
1answer
55 views

Euler number zero for odd dimensional compact manifolds

I need to prove that every compact manifold of odd dimension has Euler number zero. The Euler number of $M$ compact and oriented is $$ e(M):=\int_Ms_0^*\phi(TM) $$ where $s_0$ is the zero section of ...
3
votes
1answer
47 views

Definition of Normal Bundle and little exercise

I need to show that, given a manifold $M$ and its diagonal $\Delta\subset M\times M$, we have $T\Delta\cong\mathcal{N}_{\Delta|M\times M}$, where $T\Delta$ is the tangent bundle and $\mathcal{N}_{\...
3
votes
0answers
40 views

Euclidean Neighbourhoods Retracts and Deformation Retracts

Aim of this question is to clarify the differences between the two concepts written in the title, because it's unclear to me wether one condition implies the other, under which circumstances they are ...
4
votes
2answers
168 views

Is this CW complex a torus?

In Hatcher's book, the torus as a CW complex is constructed so: But as far as I see, I can follow the gluing instruction also in the following way. I draw the vertex $p$ and the edges $a$ and $b$ ...
4
votes
0answers
128 views

How to classify this surface

I know that it should be either a sphere, torus, Klein bottle, real projective plane, or a connect sum of any combination of these, but I don't know the steps in identifying what kind of surface this ...
2
votes
1answer
28 views

How to extend a section of fibre bundle

Let $G$ be a compact Lie group. Let $H$ be a closed subgroup of $G$, and $K$ a closed subgroup of $H$. Let $Y$ be a $K-$space. Given a map in $Map_K(H, Y)$, I wonder how to extend it to a map in $...
1
vote
0answers
34 views

Fundamental class of cotangent bundle

While reading some notes about index formula I found the expression which involved $[T^*M]$-the fundamental class of cotangent bundle. As far as I remember in order to speak about fundamental class $[...
0
votes
0answers
27 views

proof that $E(K) \leq \frac{1}{2}V(K)(V(K)-1)$

Let $K$ be a connected compact surface that has a triangulation and $V(K)$ be the number of verticals, $E(K)$ be the number of edges. Prove that: $$ E(K) \leq \frac{1}{2}V(K)(V(K)-1) $$ Thank you any ...
0
votes
0answers
59 views

What is the fundamental group of a modular curve $\mathcal{H}/\Gamma$?

Let $\Gamma$ be a finite index subgroup of $PSL_2(\mathbb{Z})$. What is the fundamental group of $\mathcal{H}/\Gamma$? By the Kurosh Subgroup theorem, $$\Gamma \cong F_n * C_2^{*r} * C_3^{*s}$$ ie, $...
3
votes
1answer
66 views

Two topological groups $\mathrm{O}(n)$ (orthogonal group) and $\mathrm{SO}(n)\times \mathbb{Z}_2$

Problem. (Basic topology (M.A.Armstrong) Exercise 16 in chapter 4.3) (1) Prove that $\mathrm{O}(n)$ is homeomorphic to $\mathrm{SO}(n)\times \mathbb{Z}_2$. (2) Are these two isomorphic as topological ...
2
votes
1answer
191 views

Problem understanding how to compute fundamental group of connected sum of torus

I have attempted trying to compute the fundamental group of a 2 torus, however I don't know how to proceed to "simplify" the result after applying van Kampen's Theorem. I calculated the fundamental ...
0
votes
1answer
41 views

Fundamental Group of the Space X

May I know what is the name of this space $X$ obtained from the picture above? Also, what is its fundamental group? I tried calculating by triangulation and using a algorithm involving maximal trees, ...
1
vote
0answers
18 views

Does every nontrivial quotient $\mathcal{H}/\Gamma$ have an unramified cover

If $\Gamma$ is a proper finite index subgroup of $PSL_2(\mathbb{Z}) \cong C_2*C_3$, then must there exist a $\Gamma'$ finite index in $\Gamma$ such that $\mathcal{H}/\Gamma'\rightarrow\mathcal{H}/\...
4
votes
1answer
78 views

How to prove that $T^1(M)$ is simply connected for some specific $M$.

Concretely, I'm working with the spaces: $S^n$, $\mathbb{C}P^n$ and $\mathbb{H}P^n$. I need to conclude that $T^1(M)$ is simply connected for all those manifolds $M$ I listed (with the exception of $...
8
votes
2answers
119 views

Existence of a simple closed curve which is not null-homotopic

Problem. Assume that $U$ is an open and connected subset of $\mathbb R^2$, and $\gamma :[0,1]\to U$ is a closed curve, which is not null-homotopic in $U$ and not necessarily simple closed. Show that ...