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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.

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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 ...
Talmsmen's user avatar
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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 $\in H_n(X)$ (...
JAG131's user avatar
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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 ...
EarnDaleheart's user avatar
2 votes
0 answers
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 ...
Talmsmen's user avatar
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2 votes
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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 ...
kaba's user avatar
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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 ...
Davi_2000's user avatar
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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} $ ...
John doe's user avatar
1 vote
0 answers
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 ...
Rick's user avatar
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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 ...
Rei Henigman's user avatar
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If $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$, Prove that $G/H \cong \mathbb{Z}$

I am trying to prove that if $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$ with both of them being free abelian groups, then $G/H \cong \...
Squirrel-Power's user avatar
2 votes
1 answer
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Proving two 2-spheres in $\mathbb{R}^{3}$ connected by line segment is simply connected.

This space is obtained by identifying the $0$ in the unit interval with the north pole of one sphere, and the point $1$ of the unit interval with the south pole of the other sphere. I am familiar with ...
JLGL's user avatar
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3 votes
1 answer
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$ S := X / \sim $. Do $S$ and$C$ have the same homotopy type?

Let $ X = \mathbb{S}^1 \times \{ \pm 1 \} \subset \mathbb{R}^3 $. We define on $ X $ the equivalence relation generated by $ (x,-1) \sim (x,+1) $ if $ x \neq 1 $, and consider the quotient $ S := X / \...
Alejandro's user avatar
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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 $\sigma, \tau$ ...
chiara casavola's user avatar
1 vote
1 answer
68 views

Question in a proof of fundamental group of Hawaiian earring is uncountable

I want to show that the fundamental group of the Hawaiian earring is uncountable. We construct the Hawaiian earring $X$ as the union of the circles $C_n$ of radius $\frac{1}{n}$ centered at $\left(\...
strugglingStudent's user avatar
2 votes
1 answer
49 views

Why do maps $E \to E'$ descend to maps between the Thom space $T(E) \to T(E')$?

Let $f: E \to E'$ be a vector bundle homomorphism between $E$ and $E'$, both equipped with metrics. Why does it follow that there exists a map between the Thom spaces $T(E) = D(E) / S(E) \to T(E')$, ...
Chris's user avatar
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Intersection number and linking number of arcs

Part 1: Let S be a surface with boundary (we assume everything is oriented and compact if necessary). It's classical that we can define the intersection number of any two closed curves. Now I want to ...
Chard's user avatar
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Computations of Homotopy Groups using long exact sequence in computation of Homotopy Groups

Given a long exact sequence associated to a (co)fibration, how can one use this to compute higher Homotopy Groups? More concretely how could one use the two sequences, $\Omega E \rightarrow Fi \...
Sm465's user avatar
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0 answers
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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 ...
Isadora Vanzella's user avatar
1 vote
1 answer
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 ...
MathGeek1024's user avatar
-1 votes
1 answer
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 ...
Jule's user avatar
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0 votes
1 answer
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 ...
EEH's user avatar
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0 votes
1 answer
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 ...
Talmsmen's user avatar
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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 ...
threeautumn's user avatar
1 vote
1 answer
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 $A\to M(f)$....
YordanToshev's user avatar
2 votes
1 answer
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. $H^k(M) \times H^l(M) \rightarrow H^{k+l}(M)$. ...
threeautumn's user avatar
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0 answers
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 ...
threeautumn's user avatar
0 votes
1 answer
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 ...
Eric Ley's user avatar
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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 ...
Stephen Jiang's user avatar
0 votes
1 answer
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 ...
Talmsmen's user avatar
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0 votes
1 answer
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 $\sim$ ...
Luca T. Castrillón's user avatar
2 votes
1 answer
90 views

de Rham Cohomology of disc using Poincaré Duality

According to the Poincar'e Duality, for a compact $n$-dimensional manifold, $$H^k_{dR}(M)\simeq (H^{n-k}_{dR}(M)).$$ But this shows that for a closed $n$-disc $D^n=\{|x|\leq 1:x\in\mathbb{R}^n\}$, $$H^...
MAS's user avatar
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1 vote
0 answers
45 views

Fundamental group of covering space as the kernel of homomorphism

Consider a surjective homomorphism $\theta: Z_2 * Z_3 \to S_3$ ($S_3$ is the symmetric group on 3 objects) given by mapping the generators to elements of order 2 and 3 in $S_3$ respectively. By ...
ricci_borel's user avatar
1 vote
1 answer
62 views
+200

Bijection involving the fundamental groupoid of a manifold

Let $M$ be a smooth manifold. I read in this post, that there is a bijection between the fundamental groupoid $\Pi(M)$ and $(\tilde{M}\times \tilde{M})/\pi_1(M)$, where $\tilde{M}$ is the universal ...
Kandinskij's user avatar
  • 3,537
3 votes
1 answer
44 views

Homology of mapping telescope calculation

Studying for quals and came across this question: Let $X_n$ be formed by taking disjoint unions of $n$ cylinders ($S^1$ x $I$) say $C_1, C_2, ... C_n$, by gluing for each $k$, the $S^1$ x $\{1\}$ of $...
ricci_borel's user avatar
1 vote
0 answers
23 views

Characteristic class detecting "upward-facing" surfaces

Let $\Sigma \subseteq \mathbb{R}^3$ be a smoothly embedded compact oriented surface with boundary. Let $\vec{n}: \Sigma \rightarrow \mathbb{R}^3$ be the field of unit normal vectors associated to the ...
JMM's user avatar
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0 votes
1 answer
41 views

How to understand May's proof that counit map is a weak equivalence?

A similar question was asked about 4 years ago here, but received no answers, so I hope it is appropriate to post a new question. I am trying to read the singular homology section in May's Concise ...
Christian's user avatar
1 vote
0 answers
33 views

The universal bundle $\gamma_k\rightarrow BO_k$ is a real vector bundle

For all $n$, let $\gamma_k^n$ bet the tautological bundle over $Gr_k(\mathbb R^n)$, i.e. $$\gamma_k^n=\{(V,v):V\in Gr_k(\mathbb R^n), v\in V\}$$ This is also naturally identified with the associated ...
Chris's user avatar
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0 votes
0 answers
43 views

Recommendation to learn cohomology and homotopy theory

I learned about the fundamental group and homology theory in Lee’s intro to topological manifolds, part 2 of Munkres and some chapters of Munkres’ elements of algebraic topology. I then polished my ...
Laplace series's user avatar
1 vote
1 answer
57 views

On the topology of $BO_k$

Let $BO_k$ be the classifying space given by: $$BO_k=\varinjlim_{\mathbb N\ni n}Gr_k(\mathbb R^n)$$ I am trying to determine aspects about the topology of this space, but cannot find any sources that ...
Chris's user avatar
  • 3,177
0 votes
0 answers
33 views

Covering Map Associated to Map from Homology

Let $X_n$ be a connected $2$-dimensional manifold with nonempty boundary and set $X_n^*:=X_n-\{x_0\}$. Assume that $H^1(X^*_n;\mathbb{R})\cong \mathbb{R}$. Then since $$H^1(X_n^*;\mathbb{R}) = \text{...
Vasting's user avatar
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1 vote
1 answer
42 views

Definition of genus: cobordism or Thom space

I know two definitions of a genus: a ring homomorphism $$\Omega_*^{U}\rightarrow R$$ or a ring homomorphism $$MU_*\rightarrow R$$ where $\Omega_*^{U}$ is the complex cobordism ring, and $MU_*=\...
s.h's user avatar
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0 votes
0 answers
30 views

What is the induced smooth covering map $F_0^*E\times I\rightarrow H^*E$?

Let $F_0,F_1:M\rightarrow N$, be smoothly homotopic maps, and $E\rightarrow N$ a smooth vector bundle over $N$. Then by the homotopy lifting property there exists a bundle morphism $\tilde{H}:F_0^*E\...
Chris's user avatar
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1 vote
0 answers
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 ...
Mircea's user avatar
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1 vote
1 answer
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 ...
Chris's user avatar
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2 votes
1 answer
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?
Julián's user avatar
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7 votes
1 answer
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 $\pi_1$ ...
Julián's user avatar
  • 1,335
1 vote
0 answers
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,...
Z.N's user avatar
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1 vote
0 answers
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 ...
TaiatLyu's user avatar
  • 151
1 vote
1 answer
39 views

Characterisation of connectedness

I was studying general topology when a question came to my mind. It can be proved that, given two points of a topological space, if there exists a clopen (i.e. open and closed set) containing exactly ...
Amanda Wealth's user avatar
2 votes
1 answer
89 views

If the sum of the modulus of the coefficients of a monic polynomial is lower than 1, then the roots are lower than 1

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 ...
Julián's user avatar
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