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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2
votes
1answer
95 views

$H^*(S^1\times S^3;\mathbb{Z})$ and $H^*(S^1\vee S^3 \vee S^4;\mathbb{Z})$ are not isomorphic as rings

I'm stuck to prove that the singular cohomology groups with coefficients in $\mathbb{Z}$, $H^*(S^1\times S^3;\mathbb{Z})$ and $H^*(S^1\vee S^3 \vee S^4;\mathbb{Z})$, are not isomorphic as rings. What ...
0
votes
0answers
37 views

Help with formalisation of a reasoning about simplicial sets

We briefly mention during lectures that the homotopy relation between two maps of simplicial sets is not necessarily symmetric. So I thought for a counterexample but I don't know how to formalise it ...
0
votes
0answers
37 views

Given $(X,A)$,$(Y,B)$ such that $X/A$ and $Y/B$ are homotopy equivalent,are their relative homology groups isomorphic?

Suppose that $(X,A)$,$(Y,B)$ are pairs of topological spaces. If $X/A$ and $Y/B$ are homotopy equivalent, are $H_*(X,A;\mathbb{Z})$ isomorphic to $H_*(Y,B;\mathbb{Z})$ ?
3
votes
0answers
54 views

Simplicial homology for infinite complexes

Simplicial homology can be viewed as a covariant functor from the category of finite simplicial complexes with continuous maps over support polyhedra, to the category of sequences of abelian groups. A ...
2
votes
1answer
65 views

Is there a long exact cofiber sequence for a homotopy pushout?

Let $f:Z \to X$, $g: Z \to Y$ and let $M(f,g)$ be the corresponding mapping cylinder. Does the homotopy pushout diagram induce a long cofiber sequence? If so what does it look like?
1
vote
1answer
50 views

How does singular homology work?: $H_1(S^1)$.

This is a fundamental inquiry into the nature of singular homology. Let $\gamma$ and $\sigma$ be the following singular 1-chains on the circle: $$\gamma(t)=2\pi t,~~~~~\sigma(t)=4\pi t,$$ Now, based ...
1
vote
2answers
71 views

triviality of vector bundles with the reduced homology of base space entirely torsion

Let $\xi$ be an $n$-dimensional vector bundle over a manifold $M$ such that the reduced cohomology $\tilde H^*(M;\mathbb{Z})$ is entirely torsion (every element has finite order under addition). ...
2
votes
1answer
29 views

Hatcher example 3H.3. Local coefficients via Modules

I'm trying to understand the following example made by Hatcher at page 329 in the section "Local Coefficients via Modules" The problem arises when I start to prove that a basis for $C_n^+(X')$ is ...
2
votes
1answer
30 views

Action induces action of group ring on singular chain complex. [duplicate]

See here for a related question. Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ ...
1
vote
2answers
93 views

Equivalence between principal $ O(n) $-bundles and vector bundles

There's a well-known result (for example, Th. 14.2.7 in tom Dieck's book) that the category of principal $ \operatorname{GL}_n(\mathbb{R}) $-bundles and bundle maps is equivalent to the category of $ ...
2
votes
1answer
128 views

Homemorphism from $S^n$ to $S^n$

Let $S^n$ be the unit $n$-sphere($n\geq2$) and $X=\{a_1,...,a_k\}$, $Y=\{b_1,...,b_k\}$ be two finite subsets of $S^n$, does there exist a homemorphism $f$ from $S^n$ to $S^n$ such that $f(a_i)=b_i$ ...
0
votes
0answers
51 views

Is the unique map $\emptyset\to \Delta^0$ really a horn inclusion? [duplicate]

Is it true that $\emptyset \to \Delta^0$ should be considered a horn inclusion? This seems to imply some terrible things, unless I'm missing something. In particular, it implies that a Kan fibration $...
2
votes
1answer
38 views

An $d$-unramified covering of compact Riemann surfaces induce a (monodromy) action on $d$ letters. Is the opposite true?

Let $S_1, S$ be compact connected Riemann surfaces, $f : S_1 \rightarrow S$ be a meromorphic function of degree $d$ that branch over $B \subset S$. The unmarried covering $f : S_1 \backslash f^{-1}(B) ...
6
votes
1answer
117 views

The ring of stable homotopy groups of spheres is not noetherian

On page 22 of this thesis, it is written that $\pi_*(\Bbb{S})$ is not noetherian. After a bit of thinking and looking online, I haven't found why this is true. A graded ring is noetherian if its ...
3
votes
1answer
62 views

Homotopy between cellular maps: an additional property

Let $f,g \colon X \to Y$ two cellular maps between (say) finite CW complexes such that $f\sim g$ via the homotopy $H \colon X \times I \to Y$. Are there any results that permits to modify the ...
1
vote
0answers
37 views

Prove that there does not exist a homotopy for a space (similar to Topologist's Comb)

Suppose $X$ is the subspace of $\mathbb{R}^2$ consisting of straight-line segments joining $(0,1)$ to the points $(1/n,0)$, for $n\in\mathbb{N}$ and the segment joining $(0,1)$ and $(0,0)$. This space ...
2
votes
0answers
36 views

Why the combinatorial second Stiefel-Whitney class is a cocycle?

From the book "Spin geometry" by Lawson&Michaelson Appendix A or this literature we know that there is a nice combinatorial way to interpret the second SW class by the transition functions of a ...
1
vote
0answers
65 views

How does the fundamental group of the base space act on its universal cover?

I have a guess: Given $p : \tilde{X} \rightarrow X$, and fixing $x_0 \in X$, then $\pi_1(X, x_0)$ acts on $p^{-1}(x_0)$ in an obvious way. (Monodromy) Is this action the action that gives $X$ as a ...
3
votes
1answer
55 views

Homotopy equivalence, Eilenberg-Maclane space to connected CW complex.

Let $X$ be any connected CW complex whose only non-vanishing homotopy group is $\pi_n(X) \cong \pi$. How do I construct a homotopy equivalence $K(\pi, n) \to X$, where $K(\pi, n)$ is an Eilenberg-...
4
votes
3answers
147 views

Eilenberg-Maclane space, when can $K(\pi, 1)$ be constructed as a compact manifold?

See here. Let $\pi$ be any group. Construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. It is rarely the case that $K(\pi, 1)$...
2
votes
0answers
39 views

Are there any resolutions for $\mathbb{Z}$ over group algebra without topological <<model>>?

Let $G$ be group. Each cell partition of the universal cover of $K(G, 1)$ delivers a (projective?) resolution of $\mathbb{Z}$ over group algebra $\mathbb{Z}G$. Can one construct a pair of ...
4
votes
1answer
62 views

CW complex such that action induces action of group ring on cellular chain complex.

Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given ...
1
vote
0answers
63 views

Construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$? [closed]

Let $\pi$ be any group. How do I construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$?
1
vote
1answer
34 views

Explicit homotopy between two maps from $\mathbb{R}P^1$ to $\mathbb{R}P^2$

I am stuck with constructing an explicit homotopy between $f$ and $g$, where $f,g:\mathbb{R}P^1 \to \mathbb{R}P^2$ are the maps defined by $$f[x,y]=[x,y,0]$$ and $$g[x,y]=[x,-y,0]$$ Notation: For $x=(...
1
vote
0answers
56 views

Classification of Torus bundle over $\mathbb{S}^1$ [closed]

$T^2$ has a fixed free isometry group as $D_8$, is this true that $T^2$ bundle over $S^1$ have three type?
0
votes
2answers
76 views

Suppose $X$ is a space and $\pi_1(X, x)=\{e\}$, the trivial group. Show that there is a homotopy

The condition is that $\gamma_0,\gamma_1$ are paths in $X$ such that $\gamma_0(0)=\gamma_1(0)=x$ and $\gamma_0(1)=\gamma_1(1)=y$, then there is a homotopy $\{f_t\}_{t\in I}$ with $f_0=\gamma_0,f_1=\...
1
vote
0answers
39 views

Hatcher's explanation on deriving homology group of real projective plane

At the page 144 of Hatcher's Algebraic topology, he calculate the boundary map $d_{k} : H_{k+1}(\mathbb{R}P^{k+1},\mathbb{R}P^{k}) \to H_{k}(\mathbb{R}P^{k},\mathbb{R}P^{k-1})$as below; To ...
1
vote
0answers
32 views

The homology group of $K^3=RP^3\#RP^3$?

Is it right? $H_0\cong H_3\cong\mathbb{Z}$, $H_1\cong\mathbb{Z}_2\oplus\mathbb{Z}_2$, $H_2\cong0$.
0
votes
0answers
33 views

fundamental group of a graph and graph homomorphisms

I am having trouble understanding what is called for in the following exercise in hatcher. Construct a connected graph $X$ and maps $f,g : X \to X$ such that $fg = {\bf 1}$ but $f$ and $g$ do not ...
3
votes
1answer
67 views

Explain unoriented $S^2$ bundle over $S^1$.

In Hamilton's classification of closed 3-d nonnegative Ricci curvature manifold, unoriented $S^2$ bundle over $S^1$ is one of the possible type. Who can give me a description of it? Many thanks!
4
votes
1answer
103 views

Degree of maps and coverings

Following a recent question I had concerning degree $1$ maps from spheres, I came up with an assumption, which might either be very easily proven false, or, if not, still hasn't been answered. It goes ...
2
votes
1answer
57 views

Triangulation for a 1-manifold

I'm taking a course on Algebraic Topology and I had a question while studying. I know the answer is affirmative but I don't know why. What I want to prove is There exist a triangulation for every ...
1
vote
2answers
88 views

Cohomology result (reference request)

I want to have reference for this result. Let $G$ be a group of order $p^k$ where $k\geq0$ and $A$ be a $G$-module. If for all positive $r$ and for all subgroup $H$ of $G$, $$H^r(H,A)=0$$ then for ...
3
votes
2answers
77 views

Does There Exist an Injective Continuous Map $f:\mathbf R^3\to \mathbf R^2$.

Question. Does there exist an injective continuous map $f:\mathbf R^3\to \mathbf R^2$? I am not able to settle even the following simpler version: Does there exists a bijective continuous map $f:\...
4
votes
1answer
145 views

What is $H_*(S^m \times S^n)$, without using Künneth theorem?

As the question suggests, what is $H_*(S^m \times S^n)$ for $m \ge 1$ and $n \ge 1$? I would like to see a way without using the Künneth theorem...
5
votes
2answers
252 views

Difference between homology and integral homology?

What is integral homology? And how does it relate to homology? I can't find a good answer anywhere, so I thought I would ask here.
0
votes
1answer
215 views

triangle with edges identified

What is the space obtained by identifying the three edges of a triangle in this way: assume the vertex of the triangle is a,b and c, then we identify ab,bc and ca. Also, what is the fundamental group ...
5
votes
0answers
56 views

Connecting homomorphism is Bockstein operation, construction of natural long exact sequence.

Let $0 \to \pi \overset{f}{\to} \rho \overset{g}{\to} \sigma \to 0$ be an exact sequence of Abelian groups and let $C$ be a chain complex of flat Abelian groups. Write $H_*(C; \pi) = H_*(C \otimes \pi)...
3
votes
0answers
28 views

If $X$ is a finite CW complex, does it follow that $\chi(X) = \chi(H_*(X; k))$ for any field $k$? [closed]

If $X$ is a finite CW complex, does it follow that $\chi(X) = \chi(H_*(X; k))$ for any field $k$?
1
vote
1answer
107 views

cup-product for singular homology?

For singular cochain complexes we defined a cup-product: $$\cup: C^p(X,A;R)\otimes C^q(X,B;R)\to C^{p+q}(X,A\cup B;R)$$ $$(\alpha\cup\beta)(\sigma):=(-1)^{pq}\alpha(\sigma_{|[0,..,p]})\beta(\sigma_{|[...
0
votes
1answer
79 views

Prove that $X/G$ is Hausdorff.

Let $G$ be a finite group acting on a Hausdorff topological space $X$. Prove that $X/G$ is Hausdorff. Deduce that the projective space $P^n$ is Hausdorff for all $n$. My Try: Consider the quotient ...
1
vote
1answer
79 views

Constructing covering space of surfaces

If $S_g$ is the surface $\#_g T^2$ where $g$ is a non-negative integer, when can we construct a covering space $S_h$ of $S_g$? Each such surface is a $CW$-complex, and in a $n$-sheeted covering, each ...
2
votes
2answers
108 views

Is totally disconnected space, Hausdorff?

Recall a space is totally disconnected if the only connected subsets are singletons (one-point subsets). Is a totally disconnected space, Hausdorff? I think it is true since if $a $ and $b $ are ...
1
vote
1answer
39 views

Prove that if $G$ is finite, then any neighborhood of a $G$-invariant subset of $X$ contains a $G$- invariant neighborhood

Let $G$ be a group acting on a topological space $X$. Prove that if $G$ is finite, then any neighborhood of a $G$-invariant subset of $X$ contains a $G$-invariant neighborhood. I have no idea even ...
5
votes
3answers
116 views

Fundamental group of $X = \{(p, q)|p \neq −q\}\subset S^n \times S^n$

This problems appears in Chapter 2, exercise 3 from "A Concise Course in Algebraic Topology J. P. May" book. Let $X = \{(p, q)|p \neq −q\}\subset S^n \times S^n$. Define a map $f:S^n\to X$ by $f(p) = ...
1
vote
1answer
40 views

questions about fundamental group being abelian

Let $X$ be a topological space and let $x\in X$. Suppose that $\pi_1(X,x)$ is abelian. Now I know that this means given two closed paths $f,g\in\pi_1(X,x)$ (i.e. start/end at x), then $[f]*[g]=[g]*[...
0
votes
2answers
56 views

Homeomorphism of $\Delta^n$ into itself that switches interior points

Let $\Delta^n$ be the standard $n$-symplex. Let $x, y$ two interior points of $\Delta^n$. How can I prove that there exists an homeomorphism of $\Delta^n$ into itself that maps $x$ to $y$? I can see ...
6
votes
0answers
106 views

Homotopy classes of functions from a finite CW complex

I am given the following problem: taken $X$ finite CW complex and $Y$ a space such that for every basepoint $y \in Y$ the group $\pi_i(Y,y)$ is finite $\forall i \leq \text{dim} X$ then the set $[X, Y]...
0
votes
1answer
28 views

Cofibration and retraction

I am working on cofibrations for an assignment and I am looking in the book Topology and geometry by Bredon and he states an inclusion map is a cofibration if and only if $A\times I \cup X\times\{0\}$ ...
4
votes
1answer
87 views

Is there an odd continuous map $f:S^{2}\to GL(2,\mathbb{C})$?

Motivated by this MO question and as an attempt to a possible generalization of the Borsuk Ulam type theorems we ask: Is there an odd continuous map $f:S^{2}\to GL(2,\mathbb{C})$ (Or $GL(2n, \...