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

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1answer
36 views

Equivariant cohomology via equivariant sheaves

Ordinary cohomology of topological space $X$ are known to be the cohomology of constant sheaf. Question Is there analogous description for equivariant cohomology? More precisely. Consider category ...
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3answers
56 views

Showing $\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}/(1,-1,0)\mathbb{Z}+(0,1,1)\mathbb{Z}+(1,0,-1)\mathbb{Z}\cong \mathbb{Z}$

I have to prove that if $V_K = \{v_0, v_1, v_2\}$ and $K = \{\{v_0\}, \{v_1\}, \{v_2\}, \{v_0, v_1\}, \{v_0, v_2\}, \{v_1, v_2\}\}$ then $H_q(K, \mathbb{Z})\cong \mathbb{Z}$ for $q = 0, 1$. Already ...
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0answers
28 views

Why did $Ext$ appear to make the sequence exact after taking its dual?

The above question is about the exact sequence in the bottom of the following figure from p196 of Hatcher's text. After taking the dual of the original short exact sequence, $Ext$ comes in at the end ...
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1answer
61 views

Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...
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1answer
62 views

Interpretation for the curvature and monodromy of a connection - Reality check

Let $P \to M$ be a principal $G$-bundle with connection form $\omega \in \Omega^1(P,\mathfrak{g})$. Here are the statements I'm basing my viewpoint on: A connection is flat (vanishing ...
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1answer
43 views

Covering space action

In exercise 1.3.28 in Hatcher's Algebraic Topology, we are asked to show that for a covering space action of a group $G$ on a simply-connected space $Y$, $\pi_1(Y/G)$ is isomorphic to $G$. This is a ...
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1answer
55 views

Is $X$ a subset of $CX$?

In Spanier's, Algebraic Topology, he writes: "A topological pair $(X,A)$ consists of a topological space $X$ and a subspace $A \subset X$." In a question at the end of the section he asks a ...
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0answers
44 views

Theta-space is a deformation retraction of the doubly-punctured plane, how to find equations.

That theta space is given by $S^1\cup(0\times[-1,1]) \subset\mathbb{R}^2$ it is said that this space is a deformation retract of the doubly punctured plane, here is the explanation I found: The ...
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1answer
64 views

The fundamental group of $B^2\times S^1$

In one exercise we are supposed to find the fundamental group of $B^2\times S^1$. It is given that the fundamental group is $\mathbb{Z}$, because we can show that $S^1$ is a deformation retract of ...
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0answers
83 views

Is there an irreducible projective hypersurface such that its complement has zero Euler characteristic?

We know that, if $f=X_0X_1...X_n \in \mathbb{C}[X_0,...,X_n]$ and $Z(f)\subset \mathbb{CP}^n$, then the Euler characteristic of its complement is zero, i.e. $$ \chi(\mathbb{CP}^n\setminus Z(f))=0. $$ ...
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1answer
68 views

Fundamental Group of Torus with Axis

I am studying for my comprehensive exams and have come across the following question, which I have been struggling with: Let $T$ be the torus given by rotating the circle ...
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0answers
91 views

Cell-structure for Grassmann manifolds, is restriction homomorphism an isomorphism for $p < k$? [closed]

Is the restriction homomorphism$$i^*: H^p(G_n(\mathbb{R}^\infty)) \to H^p(G_n(\mathbb{R}^{n+k}))$$an isomorphism for $p < k$? Here, any coefficient group may be used.
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1answer
71 views

Example 2.23 in Hatcher's algebraic topology: basis of homology group

Of course, I understand the first isomorphism and second isomorphism in this example. The problem is how induction can be used. I cannot really understand the red line. I have no idea so please help ...
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1answer
54 views

Question: Corollary 2.25 in Hatcher's algebraic topology

click here to see I can prove easily in the case of $n$ is over zero but failed to prove when $n$ equals 0. I tried like below. Since $(X_a, x_a)$ is a good pair, a pair of disjoint union of $X_a$ ...
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1answer
35 views

Deck transformations and compact CW complexes

Let $X$ be a CW complex and $\widetilde{X}$ its universal cover, formed by lifting the CW structure on $X$. A finite cellular cochain, denoted $\phi$, is a cochain in ...
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1answer
28 views

Proving another space is contractible using the homotopy extension property

Question 4 in the first exercise section in Spanier's, Algebraic Topology is stated as follows: "Prove that a space $Y$ is contractible if and only if, given a pair ($X$,$A$) having the homotopy ...
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2answers
203 views

What would the fundamental group of disjoint union look like?

Fundamental group of a wedge sum is a free product of fundamental groups. Hence, $\pi_1$ maps a coproduct of topological spaces with base points to a coproduct of groups. Since disjoint ...
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0answers
82 views

Algebraic topology needed for knot theory

Both Rolfsens Knots and Links and Lickorish knot theory require some knowledge of algebraic topology, what is a resource that covers the bare minimum I need to get through either of these? I am not ...
4
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1answer
53 views

Homeomorphism between $\mathcal{C}(X,\Omega Y)$ and $\mathcal{C}(\Sigma X, Y)$

It is easy to see that there is a natural bijection between $\mathcal{C}(X,\Omega Y)$ and $\mathcal{C}(\Sigma X, Y)$, where $\Omega Y$ is the based loop space, $\Sigma X$ is reduced suspension, $X$ ...
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1answer
24 views

Prove that quotient map of polygon is surface

We often define torus or protective map by some standard gluing of edges (quotient map) of polygon. But how to prove that the image of this quotient map is indeed a 2-manifold (Hausdorff, second ...
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1answer
25 views

The boundary morphisme of the cellular complexe of the n-Torus is the zero map.

I'm trying to compute the homology of the n-dimensional torus $ T^{n} $ . More specifically it cellular homology. I'm tending to consider the following cellular structure of $ T^{n} $. And as ...
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1answer
38 views

Can the exercise be solved this alternative way?(homomorphisms of fundamental groups)

I have this exercise: Let A be a subspace of $\mathbb{R}^n$; let $h \colon (A,a_0)\rightarrow (Y,y_0)$. Show that if h is extendabe to a continuous map of $\mathbb{R}^n$ into Y, then $h_*$ is ...
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1answer
75 views

Fundamental group of sphere with 2 handles with 2 mobius bands.

Is it real to calculate?? We have a sphere with with 2 handles and with 2 glued mobius bands (red on picture). So, i think we need to use Van Kampen Theorem 2 times. 1) $ X = X_1 \cup X_2 $ , ...
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1answer
61 views

Stably trivial bundle is trivial

I have a smooth embedding $f:S^2\to \mathbb{R}^4$ and would like to show that the normal bundle $\nu\to K$ of the image $K:=f(S^2)$ is trivial. I have already shown that it is stably trivial, i.e. $ ...
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2answers
71 views

A categorical perspective on the equivalence of sheaf cohomology and Cech cohomology?

In the nLab article on cohomology, I found the following passage. One can then understand various "cohomology theories" as nothing but tools for computing $\pi_0 \mathbf{H}(X,A)$ using the known ...
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2answers
158 views

Motivation for the nLab's definition of cohomology?

I am trying to penetrate the nLab article on cohomology. I don't know anything about higher category theory, but it seems like the real content here is topological. My question has two parts. First, ...
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1answer
156 views

Vector bundle $\gamma^1$ over infinite real projective space doesn't have finite type? [closed]

Using Steifel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{RP}^\infty$ does not have finite type?
2
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1answer
41 views

If $Y$ is path-connected, then there is only one homotopy class of maps $[0,1] \to Y$

I have this exercise: If Y is path-connected, show that there is only one homotopy-class of continuous functions from $[0,1]$ to Y. My attempt: What I need to show is that if I have two ...
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3answers
131 views

Homotopy equivalent spaces have homotopy equivalent universal covers

A problem in section 1.3 of Hatcher's Algebraic Topology is Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. ...
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1answer
21 views

Monodromy action and profinite completion of fundamental group

Let $X$ be a path-connected, locally path-connected and semilocally simply connected space and $x\in X$ an arbitrary base point. Let $p: Y\rightarrow X$ be a connected, finite cover and let ...
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1answer
78 views

When does the prime spectrum deformation retract into the maximal spectrum?

For which rings is the maximal spectrum a deformation retract of the prime spectrum? For instance, a comment to the answer to this MO question mentions it's the case if we take the ring to be the ...
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1answer
48 views

How is this connection between the groups made?

I'll give some background before my question, you may skip that if you like: Lets say that you have the unit circle $S^1$. It can be proved that the map $p: \mathbb{R} \rightarrow S^1$, given by ...
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2answers
26 views

$h,h':X\to Y$ are homotopic and $k,k':Y\to Z$ are homotopic, then $k\circ h$ and $k'\circ h'$ are homotopic.

I want to show that if $h,h':X\to Y$ are homotopic and $k,k':Y\to Z$ are homotopic, then $k\circ h$ and $k'\circ h'$ are homotopic. This means that there is a continuous map $F_1:X\times I \to Y$ ...
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0answers
19 views

The topological complexity of a topological group is equal to it Lusternik-Schnirelmann category.

I'm trying to prove that the topological complexity of a topological group is equal to it Lusternik-Schnirelmann category. I proved that the TC of a Lie group is the Lusternik-Schnirelmann category of ...
4
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1answer
29 views

Deck transformations of cover of double mapping cylinder

On page 66 of Hatcher's Algebraic Topology, he discusses the universal cover of a space $X$ which is a cylinder with its edges glued to a circle by maps $z \mapsto z^m$ and $z \mapsto z^n$. He ...
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2answers
48 views

Why do we need surjectivity of f to get injectivity of $f_*$?

If we have two topological spaces X,Y and a continuous function between f, and $f(a)=b$. We get a function between the fundamental groups: $f_*: \pi(X,a)\rightarrow\pi(Y,b)$, which is given by ...
5
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1answer
27 views

Does a weak homotopy equivalence induce an equivalence of categories on the fundamental groupoids?

Let $f\colon X\rightarrow Y$ be a weak homotopy equivalence. ($\pi_0(f)$ is a bijection and $\pi_n(f,x)$ is an isomorphism for all basepoints $x\in X$ and all $n$.) It induces a functor ...
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1answer
102 views

Where is basic algebraic topology in basic algebraic geometry?

I'm a student meeting commutative algebra and algebraic geometry for the first time. The idea of studying every (commutative) ring geometrically via its spectrum (as a locally ringed space) is ...
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2answers
47 views

Homology functors preserve coproducts

I am trying to understand the proof that homology functors preserve coproducts (using Eilenberg-Steenrod Axioms) from here. Here is the definition of admissible category for homology theory Now, ...
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0answers
43 views

Topology information of the object

I would like to know how to determine fundamental group, topological and homotopical equivalence of this object. OBJECT Can we know additional information from visual representation of this object?
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1answer
90 views

Compact subset in colimit of spaces

I found at the beginning of tom Dieck's Book the following (non proved) result Suppose $X$ is the colimit of the sequence $$ X_1 \subset X_2 \subset X_3 \subset \cdots $$ Suppose points in $X_i$ ...
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1answer
31 views

Homology of CW-complexes

I am trying to show that, if $(X, A)$ is a relative CW-complex where $X^p$ denotes the p-skeleton, then $H_n(X^p, X^{p-1})$ is zero when $n\neq p$ and equal to $\bigoplus_\alpha \mathbb{Z}$, where ...
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3answers
61 views

Textbooks for axiomatic approach to algebraic topology

Are there textbooks on algebraic topology which first starts with the Eilenberg-Steenrod axioms and then derives consequences and applications directly out of the axioms. Only at the end they show ...
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2answers
141 views

Further Generalization of Jordan Curve Theorem

Recently I have read the proof of the Jordan Curve Theorem in Munkres' Topology, I wonder whether there are some generalizations and corollaries on this theorem as follows: I know any simple closed ...
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0answers
22 views

Covering space of a topological group has same isotropy subgroup for each element in the fiber

A problem in May's Algebraic Topology book (on page 33) is Suppose $p:H \to G$ is a covering map of topological groups, and let $K < H$ be its kernel. Show that $k \mapsto (g \mapsto kg)$ ...
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0answers
25 views

Direct sum of cohomology rings. How to interpret?

Consider the space $Y:= M(\mathbb{Z}_p, 2) \vee S^4)$ where $M(\mathbb{Z}_p, 2)$ is a Moore space, i.e having trivial homology groups for all $i \neq 0,2$ where $H_0(M) \cong \mathbb{Z}$ and $H_2(M) ...
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1answer
26 views

Relative version of Kunneth Theorem for $CW$ pairs.

I am working on problem 3.2.12 in Hatcher's algebraic topology which asks to show that the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times \{x_0\})$ and $S^3 \times \mathbb{C}P^{\infty}$ have ...
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2answers
49 views

$X$ is contractible if and only if it is a retract of any cone over $X$

The first exercise given in Spanier's, Algebraic Topology is: $X$ is contractible if and only if it is a retract of any cone over $X$. I have proven the first implication, however I am stuck on ...
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2answers
58 views

how to show that $S^2/\Gamma$ is not a manifold

Let $\Gamma$ be the cyclic group generated by the matrix $$\begin{pmatrix} \cos(2\pi/3) & \sin(2\pi/3) & 0 \\ -\sin(2\pi/3) & \cos(2\pi/3) & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ Show ...
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1answer
48 views

Maximal trivial subspace in vector bundles

Let X is a locally compact Hausdorff space, given an vector bundle p: E$\to$X, a subspace Y of X is called trivial (for this bundle), if we restrict this bundle over Y, it is a trivial bundle. In ...