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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The cohomology ring for disjoint union and wedge sum

I have a question concerning example 3.14 of Hatcher's Algebraic Topology. There are a few details that confuse me, so I suppose the goal of my questions is more or less clarification - I apologize if ...
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Showing $K\cong P^2\#P^2$ via homeomorphism

If we obtain Klein bottle $K$ by the usual square with one side identified in the same direction and the other side identified in the reverse direction, then this space is homeomorphic to space $P^2\#...
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Writing down homeomorphisms in a formula

Hey I am dealing with this question since some months. How rigorously do I have to prove that two "obviously homeomorphic" spaces are really homeomorphic? Example: Cup homeomorphic to a ...
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Novikov Additivity

I'm trying to read the proof of Novikov additivity from Atiyah and Singer's paper The Index of Elliptic Operators: III (Proposition 7.1). For completeness, let me reproduce the proof (modulo some ...
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Proof of Hatcher's Borsuk Ulam theorem for $S^2$

I saw in the proof the argument in Hatcher want to prove $h:[0,1]\rightarrow S^1$ is not null homotopic to a point. But why he use that complicated argument? From his defined $n:[0,1]\rightarrow S^2$, ...
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why not $F(x,0)=x_0$?

Definition of contractible space Let $F$ be the homotopy between $\mathrm{id}_X$ and $x_0$, that is $F:X\times [0,1]\to X$ is a continuous map such that $$ F(x,0)=x,\quad F(x,1)=x_0$$ for all $x\in X$...
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Is topology done rigorously nowadays?

I took a topology class with a very bad professor in my eyes. She pretended that everything is easy and obvious and did lots of mistakes and did not really care about formal proofs. So I came up with ...
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Why $S^1$ does not deformation retract to a point

I tried to use the definition of deformation retract, i.e I have a retract $r:S^1\rightarrow {x_0}$ and I define the homotopy between $id_{S^1}$ and $i\circ r$ via $F(s,t) = tx_0 + (1-t)x$. But ...
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Understanding naturality of cup product definition.

I was reading the definition of naturality from AT (on pg. 127), it says: For example, to say that the long exact sequence of a pair is natural means that for a map $f:(X, A) \rightarrow(Y, B),$ the ...
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What is a rotation number in topology in relation to complex numbers? [closed]

I believe this has something to do with algebraic cycles on complex projective planes?
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Does exist a Banach-Tarski paradoxical decomposition of the unit cube in which each piece has the Baire Property?

The Banach-Tarski Paradox for the cube states that the open unit cube in three dimensions can be decomposed into finitely many pieces, which can then, by rotation and translation, be re-assembled into ...
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Subcomplex is closed

Hatcher, p. 520 in the appendix: A subcomplex of a CW complex $X$ is subspace $A \subset X$ which is a union of cells of $X$ such that the closure of each cell in $A$ is contained in $A$ ... It is ...
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Extendability of a continuous function on Comb Space

I have been asked to prove the following - $C$ be the comb space in the Euclidean Space $R^2$. $A =$ { $(a,0) : a=0 \ or \ a= \frac 1n, for \ some \ n \in N$}. Define $f:A \to C$ by $f(a,0)=(a,1)$. ...
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A projection is a fibration

I recently learned about the concept of a Hurewicz fibration. I tried to prove that for a product of topological spaces $X \times Y$, the projection $p: X \times Y \rightarrow X$ is a Hurewicz ...
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Cup product formula.

Here is the section of a paper I was reading named "Note on Cup-Products" by I.M. James: A formula for cup-products. The cohomology theory in what follows has coefficients in the ring of ...
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Question about subgroups on the group of homeomorphisms of a topological space

Let Y be path-connected, locally path-connected, and simply connected. Let $G_1$ and $G_2$ be subgroups of Homeo(Y) defining covering space actions of Y. (this means that each point y has a ...
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Triangulation of the topologist's sine curve

Let $X$ be $\{(x,\sin{\frac{1}{x}})\mid x\in(0,1]\}$ together with the vertical set of accumulation points, the vertical line segment at $0$. This is closed subspace of $\mathbb{R}^2$. Is there a ...
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Why this group is isomorphic to $F_\infty$?

On a study on convering spaces, I have get a subgroup of the free group $F_2$ with the generators $a$ and $b$ s.t. $H=\{a^{m_1} b^{n_1}a^{m_2} b^{n_2}\cdots a^{m_k} b^{n_k}; n_1+n_2+\cdots +n_k=0\}$ ...
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$hf\alpha:I\to S^1$ is not null-homotopic

I was studying this proof, I understood everything, except the part where $hf\alpha:I\to S^1$ is not null-homotopic. We have, that $f:S^2 \to S^1$ a continous map such that $\space f(-x)=-f(x)$ $ \...
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Clarification of what is meant by loop in $S^{1}$

I am a little confused about what is meant by a loop in $S^{1}$. For example, let $f_{1}(t)= e^{2\pi it}$ defined on the interval $A= [0, 1/2]$ and $f_2 (t)=e^{2\pi i(1-t)}$ on the interval $B= [1/2, ...
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homotopy and arrival space of an homotopic function

I have an exercise to solve where the goal is to show that a space $X$ is homotopic equivalent to a point if and only if, $\forall f:X\to Y$ is nullhomotopic for any given $Y$. I think that the best ...
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1answer
45 views

Homotopy fixed points and ordinary fixed points

Currently I know nothing about homotopy fixed points except for its definition: given a $G$-space $X$, the set of homotopy fixed points is defined as the space of equivariant maps from $EG$ to $X$. I ...
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Long fiber sequence for classifying spaces

Is there a long exact sequence for classifying spaces of topological groups? I am reading the book Basic Bundle Theory and K-Cohomology Invariants (springer webpage). On page 140, there goes a ...
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1answer
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Criterion of map $M_f\to Z$ to be continuous

I was about to prove the following proposition A map $M_f\to Z$ is continuous if and only if the induced maps $X\times I\to Z$ and $Y\to Z$ are both continuous. Here, $M_f$ is a mapping cylinder of $...
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Schubert Cells and Euler Classes

So i was wondering about the following question: Suppose we have the tautological bundle $\tau$ over the Grassmanian $Gr(n,N)$ (over $\mathbb{C}$) then we may consider the natural quotient of the ...
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42 views

Compactly generated spaces as quotients of topological sums of compact Hausdorff spaces

I have two questions about Proposition 7.9.2 and Corollary 7.9.3 in tom Dieck's "Algebraic Topology". Here is the setting, taken from Tammo tom Dieck: Algebraic Topology, European ...
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Reading request: Books like Spivak's differential geometry

Are there any series in mathematics which take a simlar approach to Spivak's differential geometry in other fields? I am currently coming towards the end of the second volume and have greatly enjoyed ...
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1answer
26 views

Why is this map an identification?

Let us define $\Delta^n=\{(t_0,...,t_n) \in \mathbb{R}^{n+1} \mbox{ such that } t_i \geq 0 \mbox{ and } \sum_i t_i=1\}$ for $n \geq 0$. If $q(\mu_0,...,\mu_{n-1},t)=(t,(1-t)\mu_0,...,(1-t)\mu_{n-1})$ ...
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1answer
26 views

Difference between homotopy function and homotopic space

I'm having some troubles understanding the concept of homotopy and homotopic spaces. I understood that given two function $f,g:X\to Y$ we can say that there's an homotopy from $f$ to $g$ if $\exists$ ...
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What is the optimal dimension $k$ for which $S^2\times S^2$ into $\mathbb{R}^k$?

Clearly $k\geq4$ since $S^2\times S^2$ is a 4-manifold and $k\leq6$ since $S^2$ embeds in $\mathbb{R}^3$. Also, since $S^2\times S^2$ is compact, "invariance of domain" argument also show ...
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1answer
28 views

Persistant homology - Point data sets from images

I have been reading about topological data analysis techniques and specifically about Persistent Homology. The examples I have seen so far use point clouds as the data sets. But what if we have, say, ...
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1answer
52 views

Question on Homotopy and Homology Groups of Spaces $X$ and $X \cup_{S^{n-1}} D^n$

I have a general question about the differnce of homology and (simplicial) homotopy groups of a topological space in degrees $X$ and the space $Y := X \cup_{S^{n-1}} D^n$ obtained by glueing $D^n$-...
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Prove that the orbital projection of a $G$-space $X$ is a functional cover

What shown to follow is a reference from the text Curso de Topología by Sergey A. Antonyan. I traslated it from spanish and thus for sake of completeness I put here a link where you can find the ...
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2answers
51 views

The definition of tori - topology

The question may seem silly, but I do not find the answer. For me, the $n-$torus is "an sphere with $n$ holes". Topologically, this is called the connected sum of $n$ tori, is it right? What ...
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43 views

Is it possible that a surjective map on a non contractible space be null-homotopic?

Let $X$ be a non contractible space and $f:Y\to X$ a surjective map. Is it possible that one have $f$ null-homotopic? If yes, Is there some result to help to prove that some specific surjective map ...
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1answer
28 views

decomposition of order preseving injective map $\phi:[n-k] \to [n]$

Let $d^j:[n-1]\to [n]$ be the order preserving injection that skip $\{j\}$ in the range,where $[n] = \{0,...,n\}$. Prove the following two fact: Let $\phi:[n-k] \to [n]$ is the injective order ...
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The fat realization of simplicial trivial $G$-bundle is principal G-bundle?

When I read Johan L.Dupont's book 'Curvature and characteristic classes', I couldn't understand his following claim. Claim Let $ \pi :E\rightarrow X$ be a principal G bundle and let $\mathfrak{U} =\{...
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Some confuse in spectral sequence and its calculate

We have the Leray's theorem: Let $\pi:E\longrightarrow B$ be a fiber bundle with fiber $F$ over a simply connected base space $B$. Assume that in every dimension $n$, $H^{\ast}(F)$ is of finite rank ...
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Understanding Hatcher's Proposition 1.26

First, let me state the proposition in Hatcher's textbook (a) If $Y$ is obtained from $X$ by attaching 2-cells as described above, then the inclusion $X\hookrightarrow Y$ induces a surjection $\pi_1(...
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Reformulating homology theory from pairs to single spaces

I want to convert chapter IX of Eilenberg-Steenrod book to an exposition in single spaces, reminiscent to usual introductory expositions of simplicial or singular homology, but for Cech homology. This ...
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On Homotopy Equivalence

If a space X is homotopic equivalent to Y and X is contractible, prove that Y is contractible I know that since X is contractible the identity map is nullhomotopic and since X,Y are homotopic ...
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Homotopy between maps, Rotman Algebraic topology

Let $x_0,x_1$ belong to $X$ and let $f_i : X \to X$ for $i=0,1$ denote the constant map at $x_i$. Prove that $f_0$ and $f_1$ are homotopic if and only if there is a continuous function $F : I \to X$ ...
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Strong deformation retraction.

Suppose $X$ is a path connected contractible space. So we know there is a homotopy $F:X\times I \rightarrow X$ with $F(x,0)=x$ and $F(x,1)=c$, for all $x\in X$ and some fixed $c\in X$. Since $X$ is ...
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Open in which topology? Seifert–Van Kampen theorem

Is a particular case of Seifert-Van Kampen theorem that if $U$ and $V$ are open sets simply connected and $U\cap V$ is path connected, then $X=U\cup V$ is simply connected. I have dificulties with the ...
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1answer
59 views

Examples of the difference between Topological Spaces and Condensed Sets

There is apparently cutting-edge research by Dustin Clausen & Peter Scholze (and probably others) under the name Condensed Mathematics, which is meant to show that the notion of Topological Space ...
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Fundamental group of 2-Tori

It is well-known that $\pi_1(T\sharp T) = \langle x_1, y_1, x_2, y_2 \mid x_iy_ix_i^{-1}y_i^{-1}\rangle$. 2-Tori has two holes and consider taking a loop between those two holes. (Between two holes ...
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1answer
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Deformation Retract of $I^2 - \{c_0\}$

Consider $I^2 \setminus \{c_0\}$, unit square with the center point removed. Call this space $X$. Consider $I^2$ with a small open disk removed about $c_0$ Call this space $Y$. Instead of taking open ...
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Characterizing simply connected spaces

A topological space $X$ is simply connected if it is pathwise connected and each closed path $u : I = [0,1] \to X$ is path homotopic to the constant path at $x_0 = u(0) = u(1)$. Recall that A closed ...
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Cell structure on the Klein bottle in $\Bbb R^3$ with deleted open disk bounded by the circle of self-intersection.

Denote $Y$ the space I mentioned in the title. The original problem is Hatcher's Exercise 1.2.12 part (b) which ask to prove $\pi_1(Y) = \langle a,b,c|aba^{-1}b^{-1}cb^\epsilon c^{-1}\rangle$ for $\...
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1answer
37 views

Smoothing a collar twist.

I am currently reading an article https://arxiv.org/pdf/1802.02609.pdf, and a question arose. The setting is at the beginning of section 2. Let $M$ be a closed, oriented, smooth $d$-manifold, and fix ...

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