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

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Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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1answer
25 views

partial converse of existence of covering spaces

suppose $X$ and $Y$ are two spaces and let $Z$ be a covering spaces for both $X$ and $Y$, then is it true that there is some space $W$ such that $X$ and $Y$ is a covering space of $W$??? (sorry but I ...
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30 views

integral cohomology ring of real projective space

What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher's book and Proving that the cohomology ring ...
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18 views

integral cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
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21 views

Definition of a simplicial complex

In the book "Algebraic Topology" by E Spanier, a simplicial complex is define a follows - D1 - A simplicial complex $K$ consists of a set $\{v\}$ of vertices and a set $\{s\}$ of non empty subsets ...
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The tangent bundle of $\mathbb{CP}^1$ is not isomorphic to its dual.

This question is related to this one but is not a duplicate. In " Characteristic Classes" by Milnor and Stasheff on pages 167-168, the authors give a brief argument about why the tangent bundle of the ...
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1answer
45 views

Which maps on the zeroth homology do actually come from continuous maps?

Let $X, Y$ be topological spaces. What are the possible maps $H_0(X) → H_0(Y)$ on homology coming from continuous maps $X → Y$? For example, can a map $X → X$ on a connected space induce a ...
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Surjective map from orientable covering space to orientation cover of base space.

Let $p \colon M \to N$ be a covering space, and let $M,N$ be manifolds. Assume now that $M$ is orientable and $N$ is not orientable. I'm asked to find a covering map $q \colon M \to N$ s.t. $p= \pi ...
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1answer
31 views

A pushout of a homotopy equivalence along

Can anybody show me an example which prove that: A pushout of a homotopy equivalence along a arbitrary map (in Top) doesn't have to be a homotopy equivalence. I know that if we change "arbitrary ...
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1answer
51 views

A problem about homotopy equivalence in Hatcher's book

In reading the proof of corollary 0.21 of Hatcher's algebraic toplology. I can not understand how the existence of homotopy equivalence $f:X\rightarrow Y$ implies that the inclusion $X\rightarrow ...
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1answer
23 views

A space homotopy equivalent to its subspace implies the inclusion map is a homotopy equivalence?

I find it not easy to understand the proof of corollary 0.21 of Hatcher's algebraic toplology. If the question of the title is true, I can understand it. But I don't know how to prove it. Can somebody ...
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15 views

homotopy between continuous functions to an absolute retract

I have the following statement to prove as one of the "fundamental" questions our topology professor wants us to know for his final: Let $X$ be a topological space, and let $A$ be an absolute ...
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2answers
51 views

What does it mean to attach a cell to a space by a map?

I am starting to study for my algebraic toplogy exam and a lot of the problems sound like: let $Y_p$ be the space obtained by attaching an $(n+1)$-cell to $S_n$ by a map of degree $p$, where $p$ ...
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0answers
66 views

Exercise 3.4 in Rotman's An Introduction to Algebraic Topology

I am self-learning algebraic topology by reading Rotman's An Introduction to Algebraic Topology. I am stuck on Exercise 3.4 on page 41. I'd be grateful for any hints or solution. Exercise 3.4: Let ...
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3answers
55 views

Is such a map always null-homotopic?

Let $X,Y$ be CW-complexes with $X$ finite dimensional and $X = \bigcup_{n \in \Bbb N} X_n$ where the $X_n\subset X_{n+1}$ are finite sub-complexes of $X$. If $f: X \rightarrow Y$, with $f|_{X_n}$ ...
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25 views

Why use class multiplication in Homotopy groups?

This is a related to a physics question Why use class multiplication to describe topological entangling and merging?. In physics, the homotopy theory is used to describing topological defects in order ...
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25 views

Computing $\pi_1(\text{Pr}(S),\mathbb{P}_0)$

Let $(S,d)$ be a complete separable metric space, and consider the space $\text{Pr}(S)$ of probability measures on $S$ that are defined on Borel sets arising from the metric $d$. Now endow ...
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1answer
31 views

Deforming $\text{id}: S^1 \to S^1$ to the symmetry $S^1 \to S^1$ such that $x \mapsto -x$

I am trying to find a deformation retraction of $\text{id}: S^1 \to S^1$ to the symmetry $S^1 \to S^1$ such that $x \mapsto -x$. I guess this deformation of maps has to respect all homotopy rules, ...
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0answers
15 views

Reference Request: James reduced product

I would like to quickly learn the basics of James reduced product (also called James construction). Anyone know some suitable material for beginners?
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19 views

Citation: earliest incidence of the Borel localization theorem

The Borel localization theorem in (Borel) equivariant cohomology states that if $T$ is a torus and $M$ a smooth $T$-manifold, with fixed point set $M^T$, then upon localizing the coefficient ring ...
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1answer
26 views

Can we deform continuously $\text{id}:\mathbb{R}^2 \to \mathbb{R}^2$ into the constant map $\mathbb{R}^2 \to \mathbb{R}^2?$

Can we deform continuously $\text{id}:\mathbb{R}^2 \to \mathbb{R}^2$ where $(x,y) \mapsto (x,y) $, into the constant map $\mathbb{R}^2 \to \mathbb{R}^2$ (where I guess $(x,y)\mapsto (0,0)\,\,$)?$ I ...
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1answer
37 views

Attaching maps in a product cell complex

This is a follow-up to those two questions: Cartesian product of two CW-complexes, and Product of CW complexes question. Consider two cell complexes $A$ (with cells $e^m_\alpha$ and attaching maps ...
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3answers
94 views

Is reduced homology a full functor on connected spaces?

Let $X$ and $Y$ be connected topological spaces. Can we then realize each map between their homologies as coming from a continuous map between their spaces? For any arrow $φ\colon \tilde H_•(X) → ...
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1answer
74 views

A question about CW complex

Given a space $X$ and a collection of subspaces $X_\alpha$ whose union is $X$, these subspaces generate a possibly finer topology on $X$ by defining a set $A\subset X$ to be open iff $A\cap X_\alpha$ ...
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0answers
45 views

how to prove that two sets have the same homotopy type

I want to prove that two bounded sets $A$ and $B$ of $R^2$ have the same Euler characteristic (number of connected components minus number of connected components of the complementary) but I think ...
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0answers
37 views

Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration

I try to understand a proof in More Concise Algebraic Topology: Localization, completions and model categories by May & Ponto (pdf). The proof is on page 262, and it is for the statement Any ...
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2answers
33 views

Converse of the path lifting lemma

We know that covering spaces have the path lifting property i.e. if $p:E \rightarrow B$ is a covering map and $u:I \rightarrow B$ is a path wish intial point $a$, then for each $w \in p^{-1}(a)$, ...
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2answers
19 views

minimal genus of surface representing a homology class

Can you give an example of a 4-manifold with embedded surfaces of different genera representing the same homology class? (If $i:\Sigma\to X$ is an embedding of a closed surface in a closed smooth ...
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0answers
44 views

Tensoring two short exact sequences

Let $R$ be a commutative ring with $1$ and consider the following short exact sequences of $R$-modules \begin{align} &0 \to M' \to M \stackrel{f}{\to} M'' {\to} 0 \qquad \text{and } \\ &0 \to ...
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Inflating open sets up to homotopy through CW skeletons

In preparation for an upcoming exam, I’m working through old exercises. Let $X$ be a CW complex with skeletons $X^0 ⊂ X^1 ⊂ … ⊂ X$. Show that for any open $U ⊂ X^{n-1}$ there is an open $V ⊂ ...
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0answers
26 views

Homology of Subspace vs. Homology of Ambient Space.

Let $M$ be a manifold embedded in $\mathbb R^n$ , so that the manifold has non-trivial $k-th$ homology for some $n \geq k\geq 0$ . How do we identify the fact that while there is a non-trivial cycle ...
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1answer
29 views

Universal Cover of the Punctured Torus

I am currently trying to compute the universal cover of the punctured torus. I originally thought it would be a lattice, but while computing the fundamental group I saw that the punctured torus is ...
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2answers
57 views

The quotient map and isomorphism of cohomology groups

Let $X$ be a closed $n$-manifold, $B$ an open $n$-disc in $M$. Suppose $p:X\rightarrow X/(X-B)$ is a quotient map. Notice that $X/(X-B)$ is homeomorphic to the sphere $\mathbb{S}^n$. My question is ...
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1answer
21 views

If the fibers of a quotient map are all discrete, is this map a covering map?

If $p:\tilde{X}\rightarrow X$ is a covering projection then I know that for every point $x \in X$ the fibre above $x$, i.e $p^{-1}(x)$, has the discrete topology. Here $p$ being a covering map means ...
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0answers
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co H-space on CW-complex [on hold]

Does someone have any idea how to prove these? If X is (n-1)-connected CW-complex of dimension <2n then there exists homotopy co-multiplication on X. If X is (n-1)-connected CW-complex and ...
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2answers
31 views
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Homology groups of orientable surfaces.

Edit: I have a proof here but when I spoke last with my professor, she told me something was close, but not quite. Can someone help me patch this proof? I've been trying to get this down for quite a ...
2
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1answer
15 views

Homotopically equivalent to Čech nerve?

I see a theorem without proof on Gelfand & Manin: Suppose $\mathfrak U=\{U_\alpha\}_\alpha$ is a locally finite open covering of the topological space $X$ such that each finite intersection ...
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2answers
42 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
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1answer
44 views

Books or texts on singularity theory

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
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1answer
19 views

Boundary maps of the projective plane as a $\Delta$-complex (homology)

Hi, very simple question here. In Hatcher's 'Algebraic Topology' the diagram above is used to describe the projective plane as a $\Delta$-complex(see p.102). Later the 2-boundary maps are given by ...
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1answer
37 views

Fundamental group of two circles joined by an arc

What is the fundamental group of two circles joined by an arc? In other words, let $S_1$ and $S_2$ be two standard circles. Let $p_1$ and $p_2$ be two points in $S_1$ and $S_2$ respectively. Join ...
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1answer
34 views

Is the stable homotopy group of sphere a commutative ring? If not, are there easy examples?

Is the stable homotopy group of spheres a commutative ring? If not, are there easy examples? In the Adams spectral sequence converging to the stable homotopy group of spheres, it seems that any page ...
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9answers
945 views

Surprising applications of topology [closed]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
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2answers
56 views

Counter example to Mostow's rigidity theorem for 2-manifolds.

I am trying to understand a counter-example to Mostow's rigidity theorem. Here is the counter example I want to understand. Take two non-isometric octagons with the sum of interior angles equal to ...
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1answer
39 views

Poincare duality isomorphism problem in the book “characteristic classes”

This is the problem from the book, "characteristic classes" written by J.W. Milnor. [Problem 11-C] Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow ...
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1answer
26 views

The Oriented Universal Bundle in Characteristic classes by J.W. Milnor.

I have a problem to understand the section "The Oriented Universal Bundle" in the page 145 of "Characteristic classes" written by J.W. Milnor. The content in that page is like below, ...
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3answers
53 views

Universal Cover of $\mathbb{R}P^{2}$ minus a point

I've already calculated that the fundamental group of $\mathbb{R}P^{2}$ minus a point as $\mathbb{Z}$ since we can think of real projected space as an oriented unit square, and puncturing it we can ...
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23 views

Proof of the Borsuk antipodal theorem

In the proof of the Borsuk antipodal theorem about antipodal functions on the $n$-sphere one usually changes from the $\ell_2^n$-sphere to the $\ell_1^n$-sphere. Fair enough, there are homeomorphic by ...
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1answer
29 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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+50

Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...