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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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13 views

Naturality of Suspension

I have a question about an argument used in the proof of a nameless lemma introduced below. Source: P. May's "A Concise Course in Algebraic Topology"; page 98). Here the excerpt: The author claims ...
1
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0answers
28 views

Homology of Hirzebruch surfaces

Let $\mathbb{F}_n:=\mathbb{P}(\mathcal{O}(-n)\oplus\mathcal{O}(0))$ be the $n$th Hirzebruch surface, where $\mathcal{O}(k)$ is the canonical line bundle on $\mathbb{P}^1_\mathbb C$, for any $k\in\...
2
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0answers
30 views

Injection on homology dual to surjection on cohomology?

Let $f:X\to Y$ be morphism between two topological spaces. We know that $H_k(X)\xrightarrow{f_*} H_k(Y)$ is injective if $H^k(Y)\xrightarrow{f^*} H^k(X)$ is surjective. I want to know if the ...
2
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0answers
33 views

Pontryagin square: Down-to-earth computer numerical values and maps

Suppose I am just a computer, and I can only read numerical values, and I cannot read any complicated math relations. Consider the most naive simple Pontryagin square, I want to translate this to a ...
1
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1answer
30 views

$A\times Y$ deformation retract of $X\times Y$ if and only if $A$ deformation retract of $X$?

Let $X,Y$ topological spaces and $A$ subspace of $X$. I know that $A\times Y$ retract of $X\times Y$ if and only if $A$ retract of $X$. Because $r:X\times Y\to A\times Y$ retraction, then $R:X\to A$ ...
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1answer
26 views

Homotopy groups of a projective bundle

I'm learning something about Hirzebrunch surfaces from here. I'm probably missunderstanding something about homotopy groups of these complex surfaces. More precisely, in the link I provided the author ...
1
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0answers
34 views

Serre spectral sequence of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2=\mathbb{R}P^\infty$

As the title shows, we have a fibration of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2\sim\mathbb{R}P^\infty$. I am trying to check my understanding of Serre spectral sequence with ...
3
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0answers
40 views

The power and the use of homotopy pullback?

A homotopy commutative diagram $\require{AMScd}$ \begin{CD} W @>\varphi_y>> Y\\ @V\varphi_xV V @VgVV\\ X @>f>> Z, \end{CD} is called a homotopy pullback, if there ...
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0answers
22 views

Künneth theorem for compactly supported cohomology of manifold.

I know the Künneth theorem for ordinary cohomology of a manifold. Is there any version for the compactly supported cohomology (over Rational) of manifolds. If we take the product of real projective ...
6
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2answers
68 views

Degree of Map between Spheres

Let $f: S^n \to S^n$ be a continuous morphism between $n$-spheres. One knows (for example using Freudenthal's suspension thm) that for all $n \in \mathbb{N}$ holds $\pi_n(S^n) \cong \mathbb{Z}$. ...
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49 views

Question regarding notation in path homology paper

In the paper, https://www.math.uni-bielefeld.de/~grigor/quivers.pdf I have two questions regarding the definitions on Page $5$. What is $\sum c$ in Equation $3.1$ ? Is this just a new edge ...
2
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2answers
46 views

Induced homomorphism between fundamental groups

Let us consider the closed disk $\overline{\mathbb{B}(0,1)} \subsetneqq \mathbb{R}^2$. Let moreover $\mathbb{RP}^2:=\overline{\mathbb{B}(0,1)}/\sim$, where the equivalence relation $\sim$ identifies ...
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0answers
27 views

Properties of group maps from compact groups to non-compact groups

I am learning how to apply math in my research. One problem I encountered was to study the properties of group mapping from a compact group (for example, SO(3)) to a non-compact group (like (Sl(2,C)). ...
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2answers
40 views

Terminal Objects in the category $htop$

Consider the Category $C$ of topological spaces with Homotopy classes of continous maps as Morphisms. An Object $T$ is terminal if for every object $X\in C$ there exist a single morphism $X\...
4
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1answer
38 views

Homotopy pullback square and fiber sequence

If we have a homotopy pullback square $$ \begin{matrix}A &\longrightarrow{}& Y \\ \ \ \downarrow & & \ \ \downarrow \\ X &\longrightarrow{}& Z \end{matrix} $$ Question: ...
5
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1answer
32 views

Find group $G$ and action of $G$ on $\mathbb{R}^2$ such that $\mathbb{R}^2 / G \approx M \setminus \partial M$, open Mobius strip

I want to find a group $G$ and an action of $G$ on $\mathbb{R}^2$ such that $\mathbb{R}^2 / G \approx M \setminus \partial M$, where $M$ is the Mobious strip, and $\partial M$ is its boundary, a ...
3
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1answer
67 views

Are Hodge numbers topological invariants for manifolds that admit a Kähler structure?

I know that all fibers in a analytic fibration (proper, holomorpic) are homeomorphic, and if the fibers are Kählerian manifolds, then they have equal Hodge numbers. Could it happen however that a ...
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0answers
31 views

Fundamental group of the following figure: (half-full sphere with two “quotiented” holes)

Let $X$ be the following figure: I want to find the fundamental groups $\pi_{1}(X)$ and $\pi_{1}(X/\beta)$ (up to isomorphism), determine whether $B$, the part of the image including the "$\beta$-...
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0answers
18 views

Invariant lifts of a closed curve on a surface of genus > 1.

I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question : Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
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2answers
95 views

Calculating $\mathbb{R}P^1$ fundamental group.

Well, I am trying to use the fact that $S^1$'s fundamental groups is free and generated by on element ($\mathbb{Z}$), denoting $\pi_1(S^1) = \langle [\gamma] \rangle$. When $\gamma$ is a loop starts ...
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0answers
67 views
+50

Homology in all dimensions of a handle body decomposition $D^m\cup H^{\lambda_1}\cup\dots\cup H^{\lambda_N}$ for $H^{\lambda_k}$ a $\lambda_k$-handle

Let $X_k=X_{k-1}\cup_{\theta} H^{\lambda_k}$ be the $\lambda_k$-handle attached to a $\dim X_{k-1}$-manifold along the embedding map $\theta:S^{\lambda_k-1}\times D^{\dim X_{k-1}-\lambda_k}\to\partial ...
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7 views

Complex of graphs having domination number greater than $k$

I am studying discrete Morse theory and as an example, discrete Morse theory is used to obtain the homotopy type of the complex of non-connected graphs of $n$ vertices. I also read that this kind of ...
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0answers
8 views

Infinite singular point of algebraic curve [on hold]

I have an algebraic curve $-(3/2) x^2 y^2 + (1/4) y^4 - c/2 x^2 = 0$ with $c<0$. This curve has $(1,0,0)$ as an infinite singular point. Please help me to know its nature.
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1answer
55 views

Does projection of a simplicial subdivision of a simplex onto a lower-dimensional face generate a subdivision of that face?

My goal is to prove that any simplicial subdivision (or triangulation) of a simplex generates a simplicial subdivision on the faces of the original simplex. Formally, let $n,k$ be integers such that $...
1
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1answer
40 views

Construction of a monad from an operad is in the CGWH category

If $\mathcal{C}$ is an operad and if $X\in\mathcal{J}$ then $CX\in\mathcal{J}$, where $\mathcal{J}$ is the category of compactly generated weakly Hausdorff spaces well-based. I'm studying the ...
3
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1answer
26 views

Show that $H_{\widetilde {x_0}}$ is a normal subgroup of $\pi_1 (X,x_0)$.

I have come across two definitions which are as follows $:$ Definition $:$ Covering Transformation $:$ Let $(\widetilde X , p)$ be a covering space. A homeomorphism $f : \...
2
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1answer
36 views

Simplicial homology and homeomorphisms

In Hatcher's book, in the introduction page of singular homology, he mentions that "it is obvious that homeomorphic spaces have the same singular homology, in contrast to simplicial homology". However ...
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0answers
79 views

Geometric interpretation of torsion homology classes

Suppose I have a homology class $x \in H_1(M)$ which is torsion of order $k$ say. Suppose furthermore that $M$ has Dimension big enough, such that every element of $H_1$ and $H_2$ can be relalized as ...
3
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1answer
26 views

The set of homotopy classes

So I came across this exercise: Show that if $Y$ is path-connected then the set of homotopy classes $[I,Y]$ of maps of $I=[0,1]$ into Y has a single point. But I am confused here. If $Y=S^1$ then ...
0
votes
1answer
31 views

Is the Teichmuller space of a surface always a contractible CW-complex?

I'm trying to prove that the classifying space of the mapping class group of a surface $S$ with genus $g$ and $n$ boundary components is homotopy equivalent to the moduli space $\mathcal{M}_{g,n}$. ...
2
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0answers
27 views

One-degree map between manifolds with boundary

Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[...
4
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1answer
58 views

Long exact sequence of cohomology group “without” Snake lemma

Let a short exact sequence $$ 0 \to L \to M \to N \to 0 $$ is a short exact sequence of $G$-modules, then a long exact sequence is induced: $$ 0\longrightarrow L^G \longrightarrow M^G \...
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0answers
41 views

$S^2 \times S^4$ is not homotopy equivalent to $\mathbb{C}P^3$ using cohomology rings

I am trying to show that $S^2 \times S^4$ is not homotopy equivalent to $\mathbb{C}P^3$ using cohomology rings. I know that $H^*{\mathbb{C}P^3} \simeq \mathbb{Z}[\lambda]/(\lambda^4)$ as a graded ...
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0answers
13 views

n-skeleton product preserving.

It is true that the n-skeleta functor $$sk_{n}:sSet\rightarrow sSet $$ is product preserving i.e., $sk_{n}(X\times Y)$ is naturally isomorphic to $sk_{n}(X)\times sk_{n}(Y)$.
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1answer
45 views

Which open subsets of $\mathbb{R}^n$ are homeomorphic to $\mathbb{R}^n$ itself?

The motivation of this question is related to the fact that in various differencial geometry books I have seen three different criteria for chart maps. These were: If $(U,\varphi)$ is a local chart, ...
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0answers
42 views

Gaining an appreciation for homology class representatives in $\mathbb CP^n$.

Given a compact oriented submanifold $N \subset M$ one says that $N$ represents a homology class in $M$ by taking $i_*(\tau_N)$ where $i_*$ is induced by inclusion and $\tau_N$ is the fundamental ...
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0answers
30 views

Retraction $S^2 \vee S^4 \rightarrow S^2$ collapsing $S^4$ to the wedge point induces homomorphism on second cohomology.

Let $r: S^2 \vee S^4 \rightarrow S^2$ be the retraction collapsing $S^4$ to the wedge point. I am trying to show that the induced map $r^{*}: H^{2}(S^2) \rightarrow H^{2}(S^2 \vee S^4)$ on the second ...
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0answers
35 views

A 2-group $\mathbb{G}$, so that always exists $0 \to BG_b \to \mathbb{G} \to G_a \to 0?$

If $\mathbb{G}$ is a 2-group, does there always exists a short exact sequence for this $\mathbb{G}$, such that $$ 0 \to BG_b \to \mathbb{G} \to G_a \to 0? $$ where both $G_a$ and $G_b$ are nontrivial ...
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0answers
23 views

Why intermediate map in Gysin sequence is multiplication by Euler class

I am reading Bott and Tu, Differential Form in Algebraic Topology. At page 178, they constructed Gysin sequence of Sphere Bundle. I am having trouble understanding the argument, $d_{k+1}$ is ...
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0answers
15 views

Reduced cohomology relation to regular cohomology

I just wanted to double check that I'm not making a mistake here. For the $0$th cohomology group $H^{0}(X;G)$ of a space $X$, we can think of the elements as being functions $X \rightarrow G$ that are ...
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0answers
55 views

Is the surface topologically unique? [on hold]

I have been doing some geometric modeling and came across this geometric object, I was wondering if it was topologically unique or it belongs to a particular geometric class (Platonic Solids, ...
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0answers
47 views

Question about showing $\mathbb{R}P^{3}$ is not homotopy equivalent to $\mathbb{R}P^{2} \vee S^3$.

A popular exercise is to show that $\mathbb{R}P^3$ is not homotopy equivalent to $\mathbb{R}P^2 \vee S^3$. The standard way is using cup products. This has been asked several times in various places ...
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1answer
76 views

To what extent are homeomorphisms just deformations?

Background. It is often said that two spaces are homeomorphic if, roughly speaking, one space can be continuously deformed into the other without any tearing and gluing. It is then emphasized that ...
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0answers
44 views

Fiber sequence, a group and an $n$-group

Given a short exact sequence $$ 1 \to B\mathbb{Z}_2 \to \mathbb{G} \to O(n) \to 1 $$ and the fiber sequence: $$ B^2\mathbb{Z}_2 \to B\mathbb{G} \to BO(n), $$ classified by $\beta \in H^3(BO(n), \...
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1answer
26 views

Two-fold branched cover of Hexagon - Which map is meant

I am trying to understand an obvious example in a paper but do not get what is meant by: "X is a hexagon and $f:X \rightarrow \sigma^2$ is a two-fold branched cover (branched at the center of $\sigma^...
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37 views
+250

Eilenberg-Zilber Theorem Proof

Let $X, Y$ be CW complexes and the following proof (from J. P. May's "A Concise Course in Algebraic Topology"; see page 102. Here the full pdf document: https://www.maths.ed.ac.uk/~v1ranick/papers/...
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0answers
27 views

Restriction of Attatching Map for CW Complexes Closed

A CW complex $X$ is a Hausdorff space and can be interpreted as a colimit $X = colim_k X_k$ of cells $X_k $ which satisfy can inclusion relations $X_{k-1} \subset X_k$ for each $k$ and are defined ...
8
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1answer
81 views

Linking number and cup product

Let $S^p$ and $S^q$ be disjoint spheres in $\mathbb{R}^n$ with $n=p+q+1$ and let $X= \mathbb{R}^n- (S^p\cup S^q)$. By Alexander duality, their fundamental classes yield cohomology classes in $\tilde{H}...
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0answers
21 views

Recommendations for books which covers Surfaces

I'm looking for books which have a good portion on Surfaces. I know Hatcher's Algebraic Topology has a decent part where he does venture into them. But I was looking for something more, surfaces + ...
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0answers
21 views

Thom spectrum $MSpin$ and $E_2$-page for a large degree $i\geq 8$

Let the $MTH$ is Madsen-Tillman spectrum (which is a close cousin of the more usual Thom spectrum $MH$) associated to tangential structure $H$. For a computation involving no odd torsion, the Adams ...