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

learn more… | top users | synonyms (1)

3
votes
1answer
32 views

On the boundary map of a locally finite chain complex

I am just learning about locally finite homology and I'm having a bit trouble understanding some of its concepts. There doesn't seem to be a whole lot of (non-advanced) literature on this topic, so I ...
2
votes
1answer
34 views

How is this “Stiefel manifold”-like fiber bundle called? How to characterize it?

Let $M$ be a smooth manifold of dimension $n$. Consider the space $V_k(M)$ of pairs $(x,\alpha)$ where $x \in M$ and $\alpha$ is a linear embedding $\mathbb{R}^k \hookrightarrow T_x M$ (or ...
1
vote
1answer
27 views

Properly discontinuous action on homology

Let $\Gamma$ be a finite group with a properly discontinuous action on $X$. How can I show that $\Gamma$ acts on $H_k(X)_{\mathbb{Q}}$? It's not clear to me why I need to take rational coefficients.
1
vote
2answers
42 views

Is there any expression to calculate the homology groups of a quotient space?

Let $B \subset A$ where $A$ is a topological space and $A/B$ the space obtained from $A$ via collapsing $B$ to a single point. I was wondering if there is any expression for $H_k(A/B)$ in terms of ...
0
votes
1answer
52 views

What does “linear and injective on each fiber” really mean?

The question is about the proof of the following result: For a paracompact space $B$, the map $[B, \operatorname{Gr}_k] \to \operatorname{Vect}^k(B)$, $[f] \mapsto f^*(\gamma_k)$ is a bijection, ...
0
votes
0answers
28 views

Homology of SO(3)

In Tohru Eguchi, Peter B. Gilkey, and Andrew J. Hanson's review paper "Gravitation, gauge theory and differential geometry," I came across the following claim about the Homology of SO(3): I cannot ...
0
votes
1answer
35 views

Cohomology of Eilenberg Maclane space

In a book on spectral sequences that I am reading, it is stated, without proof, that $H^i(K(\mathbb{Z},2);H^0(K(\mathbb{Z},1);\mathbb{Z}))$ is isomorphic to $\mathbb{Z}$ for even $i$ and $0$ for odd ...
1
vote
1answer
28 views

Definition of orientable as given in Hatcher's Algebraic Topology

A $\textbf{local orientation}$ of a manifold $M$ at a point $x$ is a choice of generator $\mu_x$ of the infinite cyclic group $H_n(M, M- \{x\} )$. For example, in the case of $M= \mathbb{R}^n$, ...
0
votes
0answers
47 views

Deformation Retract of Complement of Two Linked Circles in $\mathbf R^3$

On pg. 47 of Hathcer's Algebraic Topology, the author discusses the fundamental group of $\mathbf R^n-(A\cup B)$, where $A$ and $B$ are circles in $\mathbf R^3$ which are linked. The author writes ...
0
votes
0answers
33 views

Topology of a specific shape

How to find topology of this shape? It's Fundamental group, homotopy type and some interesting information about it?
18
votes
4answers
1k views

Why Cohomology Groups?

Why do we need cohomology groups? Homology groups are easier to compute and given two topological spaces, there is an isomorphism in homology groups if and only if there is an isomorphism in ...
1
vote
1answer
46 views

Is Hatcher's proof of thom isomorphism theorem flawed?: I don't believe that $H^n(E,E_0)\cong H^n(R^n,R^n-0)$

Let $E$ be an oriented vector bundle over $B$, a CW complex, with fiber of dimension $n$. Let $E_0$ be $E - B\times 0$. The main theorem Hatcher uses to prove the thom isomorphism theorem is that the ...
2
votes
2answers
35 views

On the surjectivity of the Hurewicz homomorphism

The Hurewicz homomorphism is a surjective homomorphism from $\pi_n(X) \to H_n(X)$ if $\pi_{n-2}(X)=0$ according to Wikipedia. But if it is surjective then how could the following (contradiction) I ...
3
votes
2answers
76 views

Complement of the Solid Torus in $S^3$ is Again a Solid Torus

On pg. 48 of Hatcher's Algebraic Topology, the author writes that the $3$-sphere $S^3$ can be thought of as the union of two solid torus. First a formal reasoning is given which is $S^3=\partial ...
1
vote
2answers
36 views

Suspension: if $X$ is $(n-1)$-connected CW, is $SX$ $n$-connected?

If $X$ is $(n-1)$-connected CW complex, is that true that $SX$ is $n$-connected? I'm trying to understand Freudenthal Suspension Theorem on Hatcher. We define the suspension map: $\pi_i(X)\simeq ...
0
votes
2answers
45 views

How to show a straight line homotopy is continuous?

Given $f$ and $g$ continuous maps from $X$ into $\mathbb{R}^{2}$, how to show that the straight line homotopy map $F(x,t)=(1-t)f(x)+tg(x)$ is continuous?
3
votes
2answers
96 views

relations between homology and cohomology

Let $p$ be a prime number and $X$ a topological space. Are the following equivalent? (1) In the homology module $H_*(X;\mathbb{Z})$ there does not exist any element of order $p$. (2) In the ...
1
vote
0answers
37 views

Simply connected compact subsets of $\mathbb R^2$

Is every simply connected compact subset of $\mathbb R^2$ weakly contractible, i.e. all homotopy groups vanish?
0
votes
1answer
25 views

Product of nullhomologous curve with $S^{1}$ factor is still nullhomologous

Let $N$ be a nullhomologous curve in a $3$-manifold $X$. Let $S^1\times X$ be the product manifold. Why is then $S^1\times N$ nullhomologous in $S^1\times X$? PS: Is nullhomologous just a statement ...
1
vote
2answers
53 views

Universal covering group and fundamental group of $SO(n)$

The universal cover of $SO(2)$ is $\mathbb{R}$, whilst the fundamental group is $\mathbb{Z}$. That is $$ SO(2) \cong \mathrm{universal\ cover}/\pi_1 $$ Likewise, I believe that the universal cover of ...
2
votes
2answers
54 views

How can I calculate the homology group of an infinite torus using Mayer-Vietoris?

I want to calculate the (simplicial) homology of the following space using Mayer-Vietoris: I have tried to do it by cutting it along the axis and getting two subspaces homeomorphic to something ...
0
votes
1answer
34 views

If an open set is evenly covered by $p$, then its open subset is also evenly covered by $p$.

If $U$ is an open set evenly covered by $p: E\to B$ and $W$ is an open set contained in $U$, then $W$ is also evenly covered by $p$. I'm trying to prove this statement, but have a difficulty. So ...
1
vote
0answers
17 views

Computation of Maps Between Sheaves

Problem: Let $X$ be a locally compact topological space, $i:Z \hookrightarrow X$ inclusion of a closed subspace, and $j:U \hookrightarrow X$ inclusion of the complement. I want to compute: ...
3
votes
2answers
69 views

Is the pairing induced by the wedge product and integration nondegenerate on de Rham forms?

Let $M$ be a compact, oriented, smooth $n$-manifold and let $\Omega^*_{\mathrm{dR}}(M)$ be the commutative differential graded algebra of de Rham forms on $M$. We can define a pairing: \begin{align} ...
0
votes
1answer
18 views

Cohomology of complex Lie groups via compact form

Let $G$ be a compact Lie group. Let $G_{\mathbb{C}}$ be a complex Lie group such that there is inclusion $i: G \rightarrow G_{\mathbb{C}}$ of Lie groups. Moreover I require that differential of $i$ ...
2
votes
1answer
58 views

Eilenberg-MacLane Spaces $K(G,n)$: Construction!

I'm looking for an easy construction of $K(G,n)$, Eilenberg-MacLane spaces. I know I can use the Postnikov Towers for the upper part $\pi_i(X)=0$ for $i > n$. For the lower part $\pi_i(X)=0$ for ...
7
votes
2answers
321 views

Is co-cohomology the same as homology?

Suppose I have a chain complex of chains $C_n$. Then one can obtain the homology groups of this complex. Now if I choose any abelian group $G$ and I consider the cochain group $C_n^*=Hom(C_n,G)$ then ...
1
vote
0answers
36 views

Fundamental group of a circle with rational lines

Let $X$ be the subset of $\mathbb{R}^2$ given by the union of the unit circle the $y$-axis all lines through the origin with rational slopes equipped with the subspace topology. Is there a simple ...
0
votes
2answers
46 views

Is this 2-complex a $K(\pi,1)$?

Consider $CW$-complex $X$ obtained from $S^1\vee S^1$ by glueing two $2$-cells by the loops $a^5b^{-3}$ and $b^3(ab)^{-2}$. As we can see in Hatcher (p. 142), abelianisation of $\pi_1(X)$ is trivial, ...
1
vote
0answers
32 views

N-th homology group of X#Y where X and Y are connected N-varieties.

Does a formula exist that returns the $N$-th homology group of given the $N$-th homology group of $X$ and $Y$? In this case, $X$ and $Y$ are connected $N$-varieties and $X\#Y$ is their connected sum.
1
vote
1answer
44 views

$d$ operator for Mayer Vietoris sequence in De Rahm cohomology

I am currently studying De Rahm cohomology, and as knowing manifolds was not a requirement for this class, we did everything on the open sets of $\mathbb{R}^n$. I have a question for the ...
0
votes
1answer
46 views

Cartesian product of compact triangulated spaces

Let $X$ and $Y$ two compact triangulated spaces, I am trying to show that $X\times Y$ is also a compact (this is obvious) triangulated space and $$\chi(X\times Y)=\chi(X)\cdot\chi(Y)$$ Any tips on ...
0
votes
0answers
28 views

homotopy invariance homology

I'm trying to understand the proof of theorem 2.10 in Hatcher (Homotopic invariance). He starts out by the construction of the prism operator and defines a linear map $\phi:\Delta^n \rightarrow I$ ...
0
votes
0answers
41 views

Proof of Invariance of Domain Using Baire Category and Local Homology

$\newcommand{\R}{\mathbf R}$ Claim. Let $m\neq n$ be positive integers. Can there exist a bijective continuous map $f:\R^m\to \R^n$? I think the answer is no and following is my argument. Purported ...
0
votes
1answer
40 views

Choice of Fundamental Domain of Torus (Dehn Twists?)

So I would like to consider a lattice $\Lambda \subseteq \mathbb{C}$ generated by $(1,\tau)$ with $\tau$ in the upper-half complex plane. This lattice $\Lambda$ will remain fixed. If you choose the ...
6
votes
3answers
182 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
1
vote
1answer
64 views

Computation of the fundamental group of $\mathbb{C}P^n$ using induction on $n$.

Let $n\geqslant 2$, I am asked to prove the following: Proposition. $\pi_1(\mathbb{C}P^n,\cdot)$ is isomorphic to $\pi_1(\mathbb{C}P^{n-1},\cdot)$. Proof. First, let us introduce some notation: ...
2
votes
1answer
122 views

Why do these two facts imply that $S^n$ is not parallelizable for $n$ even, $n \ge 2$?

Consider the following two facts. If $S^n$ admits a vector field which is nowhere zero, the identity map of $S^n$ is homotopic to the antipodal map. For $n$ even, the antipodal map of $S^n$ is ...
3
votes
2answers
80 views

$S^n$ admitting nowhere zero vector field implies identity map of $S^n$ is homotopic to antipodal map? [closed]

If $S^n$ admits a vector field which is nowhere zero, does it follow that the identity map of $S^n$ is homotopic to the antipodal map?
1
vote
1answer
38 views

Is every compact subset of $\mathbb{R}^n$ a deformation retract of some open neighborhood?

Suppose $A \subset X=\mathbb{R}^n $ is compact. Is it necessary that $ \exists$ an open set $U \supset A$ such that $A$ is a deformation retract of $U$? If yes, is there a concrete construction of the ...
0
votes
1answer
35 views

is this map an homotopic equivalence of pairs from (disk, sphere) to the disk seen attached to a space?

Hello I was studying https://www2.warwick.ac.uk/fac/sci/maths/people/staff/vincent/cohomology.pdf On page 18 given a space $A$ and a map $f:\mathbb{S}^{n-1} \rightarrow A$ he defines the cone $X := ...
1
vote
0answers
44 views

basic intuition of fundamental groups

i have taken a course in introductory algebraic topology.i am stuck on the basic informations of a space provided by fundamental groups of the space. I know that,by fundamental group of space based ...
5
votes
1answer
76 views

Cohomology ring of $n$-torus

While developing the cup product, Hatcher gives the following example: I understand most of it, but I am having trouble understanding what he means at the end by the first two sentences in the last ...
3
votes
1answer
50 views

Long Exact sequence of Relative Homotopy Groups: examples and applications

I'm going to make a talk around higher homotopy groups, and the long exact sequence of relative homotopy groups. I would like to show some nice examples and applications of this theorem after the ...
0
votes
0answers
24 views

Homology of $T^3$ generated by three copies of $T^2$

Why is the second homology of $T^3=S^1\times S^1\times S^1$ generated by $S^1\times S^1\times \{\mathrm{pt}.\}$, $S^1\times \{\mathrm{pt}.\}\times S^1$ and $ \{\mathrm{pt}.\}\times S^1\times S^1$? ...
0
votes
1answer
55 views

Isomorphic Homology implies Isomorphic Cohomology

If two complexes have isomorphic integral homology, do the dual complexes have isomorphic integral cohomology? I can also assume that the homology, cohomology are finitely-generated abelian groups. ...
2
votes
0answers
27 views

If $p:E\to X$ universal compact covering map then any continous $f:X\to S^1$ is homotopic to a constant.

Let $p:E\to X$ be a universal covering map. Suppose that $E$ is compact and $X$ is path connected. Show that any continous $f:X\to S^1$ is homotopic to a constant. Can you give me some hints?
0
votes
1answer
38 views

exercize about the foundamental group of $\mathbb{P}^n(\mathbb{R})$

Let $p$ be a point in $\mathbb{P}^n(\mathbb{R})$ and $\Sigma$ the set containing all the projective lines passing through $p$. Given $s\in \Sigma$ we can define a continous closed path (let's say ...
0
votes
1answer
42 views

Understanding Lemma 54.2 in Munkres Topology

The image below (a Lemma and proof) is taken from Topology by James R. Munkres, 2nd Edition. Munkres Lemma 54.2 I understand the entirety of the proof up and till proving the statement: $$ F \text{ ...
2
votes
1answer
40 views

Induced maps of the circle exponents

Consider the family of circle self maps $\{f_n:S^1\to S^1:x\to x^n\}_{n\in\mathbb Z}$. How can we compute the induced maps $f_{n*}$ on the $1$-st simplical homology $H_1(S^1)$? Thought ...