# Tagged Questions

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

53 views

### Show that the homology induced homomorphism $f_*:H_3(RP^3)\rightarrow H_3(S^2\times S^1)$ is a zero map.

Let $f:\mathbb{RP}^3\rightarrow S^2\times S^1$ be a continuous map. Prove that induced map $f_*:H_3(\mathbb{RP}^3)\rightarrow H_3(S^2\times S^1)$ is a zero map. I found that the third homology of ...
26 views

### Interpretation of points in covering spaces as homotopy classes of paths [on hold]

If $p:\widetilde{X} \to X$ is a covering map, $y \in \widetilde{X}$ determines a homotopy class of paths in $X$ joining the base point $x_0$ to the point $p(y)$. But a homotopy class of paths in $X$ ...
62 views

### Fundamental group contains $\mathbb{R}$ or $\mathbb{Q}$

Is there any topological space whose fundamental group contains $\mathbb{Q}$ or $\mathbb{R}$? In case of (singular) homology or cohomology, we can change its coefficients to any abelian groups (with ...
53 views

### For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$

I am working through an old qualifying exam from another university. My course did not cover as much material as what is on this test (e.g. we did not cover cohomology). So I am just working through ...
75 views

### Why the attachment to simplices in (co)homology?

I've been thinking a bit about why we define the singular homology and cohomology groups with simplices rather than, say, cubes, and it seems to me that the elementary aspects of the theory would all ...
47 views

### Counter-example for $\tilde{H} (X/A) \cong H (X, A)$?

Yo! I was not able to find a counter-example to $$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations $A \hookrightarrow X$ (or more generally whenever $A$ is a deformation ...
54 views

### If two maps are homotopic, are the images homotopy equivalent?

My question is; If two continuous maps $f,g:X\rightarrow Y$ are homotopic, are the images $Im(f),Im(g)$ homotopy equivalent? Clearly, the converse is false. If it is false, is there any condition ...
166 views

### Broken line is NOT diffeomorphic to the real line

This is from Bredon's Topology and Geometry, page 71. This comes right after the very definition of differentiable manifold, so I think no use of tangent space or 'differential' is permitted. (Bredon ...
18 views

### Question About Covering Space Classification Theorem

I'm a bit confused by Hatchers choice of words here. He says "The main classification theorem for covering spaces says that by associating the subgroup $p_{*}(\pi_{1}(\tilde{X},\tilde{x_{0}}))$ we ...
23 views

### A doubt in Whitehead's proof about cohomology with local coefficients [on hold]

In the proof of Theorem 4.9 says that $p^*:H^n(X_n,X_{n-1};G|X_n) \to Hom(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$ has image $Hom^{\pi}(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$. ...
28 views

### Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover)

Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let $X$ be a path-connected space with $\pi := \pi_1(X,*)$ abelian and ...
17 views

### Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
95 views

### On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
15 views

### Product and Join of $G$-CW-Complexes

Given a topological group $G$ and two $G$-CW-Complexes $X$ and $Y$ I want to understand the natural CW-structure on $X\times Y$ and $X*Y$. I understand that the concepts are very similar, so I want to ...
33 views

46 views

28 views

### Universal Abelian Covering Space of genus two surface [on hold]

Let M be a surface M, i am concerned with Abelian covers. These are the covering spaces for which the deck group is Abelian. The largest such cover corresponds to the commutator subgroup of the ...
23 views

532 views

### Cellular Boundary Formula

In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta}$ where $d_{\alpha\beta}$ is ...
85 views

44 views

### Fundamental group of hole-punched torus with boundary identification

I'm trying to find the fundamental group of $Y$ obtained from the torus by removing a small disk and identifying the boundary with the torus meridian. Here's my idea. The torus has the polygon ...
31 views

### Fundamental group of $S^{1}$ unioned with its two diameters

Is my solution correct? Call $X$ the riflescope space (I made this name up). I let $p$ be the point of intersection of the two diameters, and $q$ be the right point of intersection of the horizontal ...
### bottom map of pullback square is cofibration $\Rightarrow$top map is cofibration
$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I am trying to show that for a given fibration $E \xrightarrow{p} B$, and a cell structure $\{B_n\}$ on $B$ that , that $p^{-1}B_n \to p^{-1}B_{n+1}$ is ...
The obstruction to obtaining a lifting to the total space $E$ of a Serre fibration $E \to B$ of a map $X \to B$ can be derived by assuming a CW complex structure for $X$ and examining the obstruction ...