# Questions tagged [algebraic-topology]

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

21,502 questions
Filter by
Sorted by
Tagged with
18 views

### Identify a familiar fundamental group from attaching 2-cells to $\mathbb S ^ 1 \vee \mathbb S ^ 1 \vee \mathbb S ^ 1$

Question: Let $X = \mathbb S ^ 1 \vee \mathbb S ^ 1 \mathbb S ^ 1$ be the wedge sum of three circles. We shall give the circles the labels $a$, $b$, and $c$ with orientations in the counterclockwise ...
• 1,200
34 views

### Finite Order (in Homology Group) Implies Non-Orientability? (Intuition)

The motivation for this question comes from this post ("Where does the term "torsion" in algebra come from?"). To keep it brief: Given an element of finite order $\in H_n(X)$ (...
• 317
1 vote
22 views

### Fundamental Group of a Surface of genus $g$ can be realized as a deck transformation of isometries on its universal cover.

If $\Sigma_g$ is a closed surface of genus $g\ge2$, then there exists a deck transformation of isometries on its universal cover $\mathbb H^2$ isomorphic to $\pi_1(\Sigma_g)$. I’m comfortable with the ...
34 views

### Visualize $\mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2$ as an immersed surface in $\mathbb R ^ 3$

I have a short question about Munkres chapter 74 question 4 part (b). (b) Show how to picture the $4$-fold projective plane as an immersed surface in $\mathbb R ^ 3$. In the previous part of the ...
• 1,200
37 views

### Fundamental group of a hatching

Let $X = (\mathbb{Q} \times \mathbb{R}) \cup (\mathbb{R} \times \mathbb{Q})$ and $\mathcal{T}$ be the usual subspace topology of $\mathbb{R}^2$ in $X$. Let's call this space a hatching, because it ...
• 2,159
27 views

### Existance of a covering map from four circles to a bouquet of two circles

Let $R$ be the union of the four circles in the plane of radious $1$ and centered respectively at $(-3,0), (-1,0), (1,0)$ and $(3,0)$. Also, let $X$ be the union of the two circles of radius $1$ and ...
58 views

### Question about the degree of a map

Let $f: S^1 \to S^1$ be a continuous function. If $[w] \cong 1$ generator of $\pi_1(S^1) \cong\mathbb{Z}$, $\Rightarrow$ $\deg f = (\xi \circ f_*)([w])$, where $\xi: \pi_1(S^1) \to \mathbb{Z}$ ...
1 vote
61 views

### Group cohomology: Ring structure on $H^*(C_p, \mathbb{Z})$

Ring structure on $H^*(C_p, \mathbb{Z})$. I am trying to work this out but am struggling. I know what the individual cohomology groups are. But am unsure how to prove the ring structire. We have a ...
• 401
40 views

### Is there a simple description of the total space of a principal S^1 bundle over a compact surface?

It is known that principal $S^1$-bundles over a compact surface $\Sigma_g$ are classified by their Chern classes in $H^2(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}$. When the Chern number is zero, the ...
• 1,379
1 vote
56 views

39 views

### I need the demostration of the following problem [closed]

Let $X$ be a path-connected topological space, and let $x,y \in X$ be two distinct points. We aim to prove that $\pi_1(X,x)$ is abelian if and only if $u\sigma = u\tau$ for any paths $\sigma, \tau$ ...
1 vote
68 views

45 views

### Induction on proposition 2B.6 of Hatcher's Algebraic Topology

So proposition 2B.6 states that an odd map $f:S^n\rightarrow S^n$ must have odd degree. I've understood (perhaps incorrectly) that the following diagram of short exact sequences seen here induces a ...
1 vote
122 views

### Tangent vector fields on the sphere

Let $\Omega \subset \mathbb{S}^2$ be an open set such that $\Omega \cap -\Omega =\emptyset$, where $-\Omega =\{-x\in \mathbb{S}^2: x\in \Omega\}$. Is it possible to have a continuous unit tangent ...
• 193
36 views

### Abelianizer and Van Kampen pushout as a presentation

Is there a general way to write the abelianization of a group and a pushout of fundamental groups as in Van Kampen as a presentation $\langle X|R\rangle$ for $X$ a subset of a group and $R$ a set of ...
• 21
28 views

### Conjecture: For linearly homotopic loops with different base points there exists path that induces path homotopy

Let $\alpha$ and $\beta$ be loops in a space X based at x and y respectively. Suppose there exists a linear homotopy F from $\alpha$ to $\beta$. It feels fairly extremely reasonable that: If we define ...
• 21
28 views

### Confusion about finding a covering space for $\langle a^2, (ab)^4, b^2\rangle \leq \pi_1 (\mathbb S ^ 1 \vee \mathbb S ^ 1)$ (Hatcher 1.3.12)

I am confused about the covering space of the wedge sum of two circles that corresponds to a subgroup $\langle a^2, (ab)^4, b^2\rangle$ (reminiscent of $D_4$) of the fundamental group of the base ...
• 1,200
23 views

### construct homotopy between two refinement maps, exercise 10.5 of Bott, Tu

The question is the exercise 10.5, page 111, hidden in a lemma of Bott, Tu, Differential Forms in Algebraic Topology. The background is below: Lemma 10.4.2. Given $u = \{U_{\alpha}\}$ an open cover ...
• 181
1 vote
47 views

### Cofibrant replacement in Serre-Quillen model structure on Top

In the Hurewicz-Strøm model structure on $\mathrm{Top}$ there is a very straight forward cofibrant replacement for a map $f\colon A\to X$, namely by replacing $X$ with its mapping cylinder $A\to M(f)$....
43 views

### Is cohomology group corresponding to wedge product preserved under homotopy?

The explicit question is below: For de Rham cohomology, wedge product $(u, v) \rightarrow u \wedge v$ gives a homomorphism of cohomology groups, i.e. $H^k(M) \times H^l(M) \rightarrow H^{k+l}(M)$. ...
• 181
16 views

### compute the Euler class of tautological C-bundle over $CP^1$

This might be an old question. But since I have not found an explicit answer to this question, I put the question here. The background is that we need to use a similar technique when we construct the ...
• 181
28 views

### Compute the homology of this configuration space.

Let $U\subset V$ be finite labeling sets, and $K:\mathbb S^1\to\mathbb R^3$ be a knot. Consider the configuration space with points labeled $U$ lying on the knot, to make this space connected we fix ...
• 728
41 views

### What is a direct limit of chain complexes?

As in this post, I have been stuck on exercise 3.3.17 of Hatcher. In particular, the textbook defines a 'direct limit of chain complexes', but what exactly does that mean? Would deeply appreciate any ...
62 views

### Find $\pi_1(K\#\dots\#K\#\mathbb S ^ 2)$

Find $\pi_1(K\#\dots\#K\#\mathbb S ^ 2)$ where $\#$ is the connected sum of two topological surfaces and $K$ is the Klein bottle. We induct on the number of Klein bottles considered in the connected ...
• 1,200
65 views

### Fundamental group of this space is $\mathbb{Z}/n\mathbb{Z}$

Consider $S^3$ as a topological group with the product in $\mathbb{H}$. Let $R_k$ be the set consisting of the $k$-th roots of unity in $\mathbb{C}$. Then $R_k$ is a subgroup of $S^3$. Consider $\sim$ ...
90 views

• 2,065
1 vote
42 views

• 3,177
1 vote
74 views

### A question regarding the deformation retraction of a ball without origin to a sphere.

Let $D=\{||x||\leq 1\}\subset \mathbb R^d$ be a ball or radius $1$ with center at origin $O$ and let $H:\overline (D\setminus\{O\})\times[0,1]\to D\setminus\{O\}$ be a strong deformation retraction to ...
• 554
1 vote
41 views

### Let $f,g:M\rightarrow N$ be smooth maps transverse to $X\subset N$. Are $f^{-1}(X)$ and $g^{-1}(X)$ cobordant?

Let $F,G:M\rightarrow N$ be smoothly homotopic maps so that $F$ and $G$ are transverse to $X\subset N$, are $F^{-1}(X)$ and $G^{-1}(X)$ then cobordant to one another? I saw this statement in these ...
• 3,177
47 views

### Adjunction between $Top$ and $hTop$

There's a functor $U : Top \rightarrow hTop$ that is the identity on objects and is the quotient projection (up to homotopy) on hom sets. Does this functor have a left (or right?) adjoint?
• 1,335
178 views

### Isn't the fundamental group a hom functor?

Can you not define the fundamental group as $\pi_1 := hTop_*\big((S^1,s_0),-\big)$ ? (You would need to prove it has a group structure separately.) I thought this was so, but then I saw that $\pi_1$ ...
• 1,335
1 vote
95 views

### Fundamental groups of Calabi-Yau manifolds

This may be a very basic question. Let $(X,J,g)$ be a compact Kahler manifold, where $J$ is a complex structure and $g$ is a Riemannian metric. We assume that $(X,J,g)$ is a strict Calabi-Yau $m$-fold,...
• 11
1 vote
39 views

### Decomposition theorem for resolution of surface singularities

In the section 3.1 of the paper Intersection forms,topology of maps and motives decomposition for resolution of three folds by de Cataldo and Migliorini: https://arxiv.org/abs/math/0504554 They prove ...
• 151
1 vote
Say we have a monic polynomial $$x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb C[x]$$ such that $\sum_{k=0}^{n-1} |a_k| < 1$. Then is it the case that all roots are in $B(0,1)$? This was ...