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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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1answer
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Problems understanding 0th relative cohomology

I am trying to compute $H^{i}(\mathbb{R}^2,\mathbb{R}^2-S^1)$. For that, I have used the long exact sequence of relative cohomology: $0\to H^0(\mathbb{R}^2,\mathbb{R}^2-S^1)\to H^0(\mathbb{R}^2)\to ...
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1answer
46 views

$A\otimes \mathbf{Q}\cong B\otimes \mathbf{Q}$ implies $A$ and $B$ are $\mathcal{C}$-isomorphic

I am trying to solve exercise 211 on Davis-Kirk: Let $\mathcal{C}$ be the class of torsion abelian groups. Show that for any abelian groups $A,~B$, $A\otimes \mathbf{Q}\cong B\otimes \mathbf{Q}$ ...
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0answers
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Is self intersection still homeomorphic?

I am not sure how to aptly phrase my question, but if we have a surface and we self intersect a part of it, will it still be homeomorphic to our original surface? I ask because in obtaining a Klein ...
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1answer
39 views

Homotopy classes of self-maps on $\mathbb{S}^1\vee\mathbb{S}^1$

Consider the two inclusions $\eta_i:\mathbb{S}^1\to \mathbb{S}^1\vee\mathbb{S}^1$. I claim that the following map is injective $$(\eta_1\sqcup\eta_2)^*:[\mathbb{S}^1\vee\mathbb{S}^1,\mathbb{S}^1\vee\...
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1answer
26 views

Using grid diagram to compute the Alexander polynomial

I have been reading the book 'Grid Homology for Knots and Links' (see https://web.math.princeton.edu/~petero/GridHomologyBook.pdf) - in Section 3.3 it provided a way to compute the Alexander ...
2
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1answer
56 views

Prove that $SU(n)$ is simply-connected (using Van Kampen)

I have some problems to prove that $SU(n)$ is simply-connected, for $U(n) = \{ M \in GL_n(C) \; | \; ^{t}\overline{M}M = Id \;\}$. In fact, there is some indications (to follow if it's possible). ...
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2answers
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Does the inclusion of a 1-dimensional space into another induce isomorphism on fundamental groups if the spaces have fundamental group $\mathbb{Z}$?

Let $A \subset B$ be 1-dimensional CW-complexes that both have fundamental group $\mathbb{Z}$. Does the inclusion $A \hookrightarrow B$ induce an isomorphism on fundamental groups? Corollary 3.3 from ...
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0answers
25 views

Orbit space of $\mathbb{Z}_2$-action on the torus $T^2$

Let $\mathbb{Z}_2$ be the group $\{1,-1\}$. I want to construct an action of $\mathbb{Z}_2$ on the torus $T^2$ such that the orbit space is homeomorphic to the sphere $S^2$. Could anyone give some ...
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0answers
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Weil Model of Equivariant Cohomology

I was reading this paper, and was stuck on a supposedly trivial calculation at page 13. I have trouble understanding the authors' calculation of $d_X$. The authors claimed $D\lambda a= D(a-i(X)a\...
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0answers
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Degree of Line bundle and Intersection

Let $(\Sigma,g)$ be a closed Riemann surface with metric $g$. For any holomorphic line bundle $L\to \Sigma$, given a metric we have its curtature in terms of Chern connection $A$. It is well-known ...
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1answer
39 views

if $\operatorname{Hom}(A,\mathbb{Q})=0$ and $\operatorname{Ext}(A,\mathbb{Z}_p)=\operatorname{Hom}(A,\mathbb{Z}_p)=0$ implies $A=0$

I want to prove that If $A$ is an abelian group such that $$\operatorname{Hom}(A,\mathbb{Q})=0$$ and, $$\text{Ext}(A,\mathbb{Z}_p)=\text{Hom}(A,\mathbb{Z}_p)=0 \text{ for every prime } p$$ ...
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0answers
25 views

Relationship between locally trivial and locally non vanishing section.

What I want to show is that If a line bundle (not necessarily locally trivial) has locally nonvanishing sections for each open set in the open cover of the base space, then it is locally trivial. ...
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1answer
31 views

A compact $n-1$ manifold can be embedded in $\mathbb{R}^n$ iff it can be embedded in $S^n$

I would like to ask how to show that a compact $n-1$ manifold embedded in $S^n$ can be embedded in $\mathbb{R}^n$? By Alexander duality one can show that some space can not be embedded in $\mathbb{R}^...
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1answer
25 views

Singular Cohomology satisfies the Mittag-Leffler Condition on CW complexes?

Is this true? Let $H$ be singular cohomology. On an arbitrary CW complex $X$, given a filtration $X^0 \subseteq X^1 \subseteq \cdots X^n \subseteq \cdots \subseteq X$. Then $$H^*(X) \cong \...
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1answer
42 views

Postnikov towers agreeing at some stage

Let $\cdots \to X_2\to X_1$ be a Postnikov tower for an $n$-dimensional CW-complex $X$. Given some $k<n$, is it possible to find a CW-complex $Y$, such that $\dim(Y)\leq k$ and with a Postnikov ...
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2answers
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Do i imagine the linear (straight line) homotopy in a correct way?

Today i learned about the linear homotopy which says that any two paths $f_0, f_1$ in $\mathbb{R}^n$ are homotopic via the homotopy $$ f_t(s) = (1-t)f_0(s) + tf_1(s)$$ Am i right in imagining the ...
1
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1answer
31 views

Covering map of the universal cover $\widetilde{G} \rightarrow G$ for $G$ a Lie group is a homomorphism?

In a paper I'm reading, we have a compact Lie group $G$ and he says "We can identify $\pi_1(G)$ with the kernel of $\widetilde{G} \rightarrow G$. I can't seem to find anything that says that the ...
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0answers
23 views

restriction operators and continuous maps from Hom(X,Y) to Hom(A,Y)

I am trying to solve this topological question and would like to know if I am on the right track with my solution. As in if I have the correct answer or if I need to add or delete anything. However, I ...
2
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1answer
33 views

Show that the Compact open topology on Hom(X,Y) is hausdorff

I am trying to complete this topological question and I would like to know if my solution is correct. Any help would be greatly appreciated! My Solution: Let X be a topological space, and Y a ...
2
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1answer
56 views

What is Thom Isomorphism?

I am reading the following post on Thom Isomoprhism and I also have the Thom Isomoprhism from Hatcher's, Corollary 4.9,pg441 nlab's: Let $V \rightarrow X$ be a rank $n$ vector bundle over a simply ...
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0answers
40 views

Ring structure of $\Bbb CP^n$ and Chern class.

In this notes Prop 1.71 in nlab, the author aims to compute $H^*(\Bbb C P^n, \Bbb Z)$. I have two confusions. What makes it justified to use $c_1$ as the generator of $H^2(\Bbb CP^n, \Bbb Z)$? There ...
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1answer
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Associativity of a homology ring of an H-space.

Let $X$ be a based topological space with $\mu:X\times X\rightarrow X$ which is homotopy associative and unital wrt the basepoint. We call such $(X,\mu)$ an associative H-space. The multiplication on $...
4
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1answer
53 views

Fundamental group of the complement of $3$ disjoint hypersurfaces in $\mathbb{C}P^2$

Let $X$ be the union of $3$ hypersurfaces in $\mathbb{C}P^2$, then how to compute the $\pi_1(\mathbb{C}P^2\setminus X)$? What I know is the complement of a hypersurface in $\mathbb{C}P^2$ is $\mathbb{...
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0answers
43 views

How to combine shuffles to prove associativity of Eilenberg-Zilber map

I've got a problem related to $(p,q)$-shuffles that comes from the Eilenberg-Zilber map $\nabla$ when I tried to show that this map is associative in the sense that $\nabla(\nabla\otimes 1)=\nabla(1\...
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2answers
154 views

Universal covering space of the real projective line?

I´m thinking about universal covering spaces. I´ve seen a lot of examples and authors ever say "the sphere $S^n$ is the universal covering space of the $n$-dimensional projective space $\mathbb{R}P^n$ ...
0
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1answer
16 views

Ways of proving that (Combinatorial) Fundamental Group is independent of Maximal Tree

A combinatorial definition for the fundamental group is to introduce generators $g_{ij}$ for each pair of vertices $v_i,v_j$ for which $i<j$. And $g_{ij}g_{jk}=g_{ik}$ whenever $v_i,v_j,v_k$ span a ...
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1answer
22 views

Inclusion of $n$-skeleton induces surjection of cohomology rings

Suppose $X$ is a CW complex and $X_n$ its $n$-skeleton, the cohomology ring $H^*(X)$ of $X$ is defined to be the direct sum of its cohomology groups with multiplication as cup product. Does the ...
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0answers
8 views

Independence of Vertex Ordering in (Combinatorial) Fundamental group

A combinatorial definition for the fundamental group is to introduce generators $g_{ij}$ for each pair of vertices $v_i,v_j$ for which $i<j$. And $g_{ij}g_{jk}=g_{ik}$ whenever $v_i,v_j,v_k$ span a ...
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0answers
29 views

CW Structure on Self-Homeomorphism Groups

If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space $$Map(X,X)$$ (which we furnish with the compact-open topology) ...
2
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1answer
25 views

A description of the map between Grassmanians $G_1^k \rightarrow G_k$,

We know that $G_k:=co\lim G_k(\Bbb C^n)$ is the classifying space for $k$ dimensional complex vector bundles. With total space $E_k = \{(x,v) \, :|, x \in G_k, v \in \Bbb C ^\infty \}$. So we may ...
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1answer
41 views

Free product with amalgamation vs pushout [duplicate]

As in title, in terms of group theory (I'm not familiar with category-theoretic terms), question comes from algebraic topology but seems to be of general interest. (Other questions on MSE touch on the ...
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0answers
15 views

Visualizing a 2-torus on a planar shape using periodic boundary conditions

A torus can be represented by a square with periodic boundary conditions, which makes it easy to draw embeddings of graphs on the torus using a piece of paper. Is there a similar mapping of the 2-...
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0answers
36 views

Any compact orientable surface is a branched cover of a torus

Given a torus $T$, assume all the branch points are of index $2$, then by Riemann-Hurwitz theorem, the number of branch points is $2g-2$. Select $n:=2g-2$ points $\{x_i\}$ in $T$. Does that mean we ...
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0answers
26 views

Suppose $Y_f=D^n \cup_fY$ is the space obtained from $Y$ by attaching an n-cell via the map $f: S^{n-1} \rightarrow Y$, $n \geq 3$.

Suppose $Y_f=D^n \cup_fY$ is the space obtained from $Y$ by attaching an n-cell via the map $f: S^{n-1} \rightarrow Y$, $n \geq 3$. I'm trying to understand the homology groups $H_{n+1}(Y_f),H_{n}(...
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0answers
30 views

Definition of mapping telescope

In Kochman's stable homotopy theory, pg 121 prop 4.24 We let $X$ be a based CW complex. Let $X^n$ be an increasing sequence of subcomplex whose union equals $X$. We define $$TX = \bigcup_{n \ge ...
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1answer
31 views

Fundamental group of group of homeomorphism of a compact surface

I'm reading "A Primer on Mapping Class Group", and there is something I don't understand in the proof of Theorem 4.6. Define $\mathrm{Homeo}^+(S)$ to be the group of orientation-preserving ...
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0answers
30 views

Why is $Hom(T(-1),O(1)) \cong \Lambda^2(\mathbb{C}^{n+1})^*$ on $\mathbb{CP}^n$?

I am currently trying to read a paper and the author claims the following. On $\mathbb{CP}^5$ we have $$Hom(T(-1),O(1)) \cong \Lambda^2(\mathbb{C}^{6})^*.$$ The proof is claimed to be a consequence of ...
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1answer
76 views

Serre Spectral Sequence I

I am trying to follow the proof in Kochman's Introduction to Stable Homotopy Theory, page 59. This will be first part of a series of post. (Serre Spectral Sequence) Let $R$ be a commutative ring ...
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On proving that if $X,Y$ are CW-complexes and $f : A \to Y$ is continuous with $A \leq X$, $f(A^n) \subset Y^n$, then $X \cup_f Y$ is a CW-complex.

As the title says, I have a question regarding the following exercise, Let $A,X,Y$ be CW-complexes with $A$ a subcomplex of $X$. If we have $f : A \to Y$ such that $f(A^{(n)}) \subset Y^{(n)}$ for ...
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1answer
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Prerequisite of understanding a topological construction in Spectral Sequences

This is a post on the construction of a spectral sequence. I am in fact lost in the first paragraph. Let $B$ be a CW complex and $\pi\colon X\to B$ a Serre fibration. Put $X^k=\pi^{-1}(B^k)$. A ...
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1answer
33 views

What are the maps in the long exact sequence of homotopy groups for the free loop space fibration?

Let $(X,x)$ be a pointed connected CW complex, and let $\mathcal L X = Map(S^1, X)$ be its free loop space. We have a fibration $\mathcal L X \to X$ given by evaluating at the basepoint $0 \in S^1$, ...
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0answers
17 views

Reduced VS non-reduced cohomology ring

Assume cohomology ring of topological space $X$ has the form: $H^{*}(X) = \mathbb{Z}[x]/(x^k)$, where $x \in H^{1}(X)$ is a generator. What can I say of its reduced cohomology ring $\tilde{H}^{*}(X)$...
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1answer
33 views

Using Van Kampen's Theorem to determine fundamental group

I'm trying to calculate the fundamental group of a surface using (i) deformation retracts and (ii) Van Kampen's Theorem. I'm really struggling understanding the group theory behind it and the ...
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1answer
53 views

Connected boundary implies $\pi_1(M,\partial M)=0$.

I have two questions: Let $M$ be a compact connected manifold with boundary. 1, If the boundary $\partial\tilde{M} $ of universal covering $\tilde{M}$ is connected, is $\partial M$ connected? How ...
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1answer
136 views

Research level algebraic topology

I became very interested in Algebraic Topology more or less recently, and learnt a lot of "classical algebraic topology", including : Hatchers' Book, and more categorical approaches here and there ...
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2answers
43 views

Compute the effect of $∆_∗$ on homology groups.

Let $∆ : S^n → S^n × S^n$ be the diagonal map $∆(x)=(x,x)$. Compute the effect of $∆_∗$ on homology groups. My attempt : I considered the CW chain complex of $S^n$ and let, $H_0(S^n)=\Bbb Z\{v\}$ ...
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1answer
49 views

Prove that there is a long exact sequence $\ldots → H_k(X) \overset{2}\to H_k(X) → H_k(X;\Bbb Z/2) → H_ {k−1} (X) → · · ·$

Consider the exact sequence $0 \to \Bbb Z \overset{2}{\to} \Bbb Z → \Bbb Z/2 → 0$. Prove that there is a long exact sequence $\ldots → H_k(X) \overset{2}{\to} H_k(X) → H_k(X; \Bbb Z/2) → H_ {k−1}...
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0answers
28 views

Connected sum of Torus and Klein bottle.

If we look at the polygons of these spaces and we cut out a piece and glue it together like so, Do we lose any information when we compute the fundamental group using CW complex method described in ...
0
votes
2answers
43 views

Prove that any map $T → T$ ($T$ = Torus) whose restriction to $S^1 ∨ S^1$ is null-homotopic induces a $0$ map on reduced homology

Prove that any map $T → T$ ($T$ = Torus) whose restriction to $S^1 ∨ S^1$ is null-homotopic induces a $0$ map on reduced homology. A few informations which I know : (1) $T$ is obtained by attaching ...
2
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0answers
23 views

Any map from $X \to {(S^1 )}^k$ induces a 0 map on reduced homology.

Let $X$ be a simply connected space. Prove that any map from $X \to {(S^1 )}^k$ induces a 0 map on reduced homology. My attempt: Let $f : X \to {(S^1 )}^k$ be a map. $X$ simply connected implies ...