Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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11 views

Properties of a basic construction on co-H-spaces

Suppose $Y$ is a co-H-space. If $Y$ is standard, in the sense of Iwase, then $Y$ admits a homotopy decomposition $Y\simeq X\vee S$ where $X$ is simply-connected and $S$ is a wedge of circles. If $Y$ ...
4
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1answer
47 views

Definition of the infinite unitary group

I'm studying Suslin's proof of Bott Periodicity for the infinite unitary group $U$ which I currently understand to be $$\bigcup_{n\in\mathbb{N}}U(n)$$ where U(n) denotes the group of $n\times n$ ...
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1answer
35 views

How to show there is no homotopy between two curves?

On $\mathbb{C}\setminus\{0\}$, it seems that there can't be a homotopy between the curve given by $e^{i\theta},0\leq\theta<2\pi $ and the curve given by $e^{i\theta}+10,0\leq\theta<2\pi $. But ...
3
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1answer
33 views

Geometric intuition for homotopy invariance of fiber bundles?

There's a nice result in algebraic topology saying that given a fiber bundle, its pullbacks along homotopic maps are isomorphic as bundles. Thinking of a bundle as a comb with the "teeth" as its ...
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1answer
77 views

Fiber bundles that can be turned into a fibration that is a fiber bundle.

Let me recall a standard construction. Up to homotopy equivalence, any map $f: X \to Y$ is a fibering. Take the special case where $X=E$ the total space of a fiber bundle, and $Y$=B, the basespace ...
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31 views

When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$.

Let $X$ be a connected CW complex. Let $X_n$ fit into a commutative postnikov diagram for $X$ and let the fibrations $K(\pi_n(X),n) \hookrightarrow X_n \xrightarrow{\mathscr p} X_{n-1}$ be given. ...
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19 views

Minimal axioms for $X_n$ to be nth element of postnikov tower

Mosher and Tangora motivate using postnikov towers by saying they want a space $X(n)$ that represents in a certain sense, $X$, through dimension $n$. I am teaching a lecture on postnikov towers ...
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54 views

$S^n/S^k$ is homotopy equivalent to $S^n\vee S^{k+1}$.

How can I prove that $ S^{n}/S^{k} $ is homotopy equivalent to $S^{n} \vee S^{k+1} $? Here $S^{n} \vee S^{k+1} $ is a space $ \left[ S^{n} \cup S^{k+1} \right]/\approx $ where $ \approx$ is an ...
4
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1answer
69 views

Calculate $\pi_2(S^2 \vee S^1)$

I am trying to calculate $\pi_2(S^2 \vee S^1)$ and having trouble fitting the pieces together. I know that the universal cover of $S^2 \vee S^1$ is just $\mathbb{R}$ with spheres attached at integral ...
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1answer
51 views

Hatcher's proof of Van Kampen Theorem

I am studying Van Kampen Theorem using Hatcher's textbook. I am dealing with the general statement, I mean: (pg 43) He defines previously the free product of groups (pg 41) as: I can follow the ...
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1answer
25 views

In R2\(axis X)the two nullhomotopic functions are not homotopic

I want to prove that in space R2\X the two functions f&g are not homotopic if we define g(x)=c1& f(x)=c2 c1 and c2 are two points one of the above axis X and the othe one is under it
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1answer
50 views

$C_\infty$ analog of the correspondence between $A_\infty$-alg. structures on $A$ and dg coalg. strucures on $(\bar T(sA),\Delta)$

There is a 1-1-correspondence between $A_\infty$-algebra structures on a graded vector space $A$ and dg. coalgebra structures on the bar construction $(\bar T(sA),\Delta)$. My question: Is there any ...
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1answer
54 views

about properties of homeomorphism

we know that the following shapes are homeomorphic but in my book write : Geometrically speaking, a homeomorphism is a bijection that can bend, twist, stretch, and wrinkle the space M to make it ...
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1answer
53 views

Universal covering space VS fibration from contractible total space

For a path-connected space $X$, a covering space is a fiber bundle with a discrete set. It is known that if $X$ in addition locally path-connected and semilocally simply-connected, then $X$ has a ...
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1answer
46 views

Multiplicative structure on algebraic K-theory

Let $R$ be a commutative ring. Using Quillen's $+$-construction, it is relatively easy to see that the algebraic K-theory of $R$, $K_*(R)$, admits a graded commutative product $$K_i(R)\otimes K_j(R) ...
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27 views

Spectrum of spectrum in the stable homotopy category

Let $\mathcal{E}=(E_0,E_1,\cdots)$ be an $S^1$-spectrum. Define $\Sigma \mathcal{E}$ to be the spectrum with $(\Sigma \mathcal{E})_n=E_{n+1}$. Then, consider the spectrum $\tilde{\mathcal{E}} ...
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41 views

Expositions of Postkinov Towers

I am giving a lecture on Postkinov towers and I want to teach the students a lesson :). These students have seen local coefficients, spectral sequences and cohomology operations at the level of ...
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1answer
33 views

$1_{S^{n-1}} \simeq$ to a constant map

I have to show that $ \exists\ \ f : D^n \rightarrow S^{n-1} $ with $f\circ i =1_{S^{n-1}} \iff 1_{S^{n-1}} $ is homotopic to a constant map. I don't know how to prove this. So, please help me in ...
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31 views

What does formality of a chain complex mean topologically?

I've been told that every topological abelian group is a product of Eilenberg-Mac Lane spaces, but I don't have a reference for this fact. This confuses me because via the Dold-Kan correspondence, ...
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1answer
29 views

Define a map $\Omega \Sigma \Omega Y \to \Omega Y$

Let $Z$ be $\Omega Y$ which is the space of loops based at $Y_0$. Then I know how to define a map explicitly from $Z \to \Omega \Sigma X$. It is defined by noting we have the identity map $ \Sigma ...
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2answers
41 views

Show that $f$ is homotopic to $g$. [closed]

Let $X$ be any topological space. If $f,g:X\to S^n$ (n-sphere) are continuous, such that $f(x)$ and $g(x)$ are never antipodal, show that $f$ is homotopic to $g$. I have no idea about this, ...
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78 views

Fibration: if map $i^*: H^*(X, G) \to H^*(F, G)$ is surjective, then action of $\pi_1(B)$ on $H^*(F, G)$ trivial?

For a fibration $F \overset{i}{\to} X \overset{p}{\to} B$ with $B$ path-connected, if the map $i^*: H^*(X, G) \to H^*(F, G)$ is surjective, then does it necessarily follow that the action of ...
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1answer
47 views

Homotopy category of a simplicial category

In many places (for example here) I've seen the following definition: For a simplicial category $\mathcal{C}$, it's homotopy category is defined to be the category $Ho(\mathcal{C})$ with the same ...
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1answer
44 views

Removing cells from CW complex and without changing the homotopy type

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
2
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2answers
60 views

CW complex structure for $\mathbb{R}P^n$ or a space homotopy equivalent

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
2
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2answers
36 views

Are there any nontrivial second Hurewicz homomorphisms for familiar compact 6-dimensional manifolds?

Based on several computations I have done, it seems that the second Hurewicz homomorphism $$h:\pi_{2}(X)\rightarrow H_{2}(X;\mathbf{Z})$$ has a habit of being trivial. For instance, this seems to be ...
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1answer
24 views

Converse of realisation lemma for bisimplicial sets

Given two bisimplicial sets $X_{\bullet,\bullet}$ and $Y_{\bullet,\bullet}$, we have the result that if given a map $f:X_{\bullet,\bullet}\to Y_{\bullet,\bullet}$ such that the restriction ...
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1answer
45 views

Do path homotopy classes of concatenated paths have a middle fixed point?

If $[a]$ and $[b]$ are path homotopy classes, then $[a]\cdot[b]$ is defined as $[a*b]$, where $a*b$ is defined as the concatenation of the paths $a$ and $b$. Let us say $a(1)=b(0)=p$. Then does each ...
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1answer
25 views

How does one compute the Hurewicz homomorphism for a (symplectic) nilmanifold?

I have a symplectic six-dimensional nilmanifold $X:=G/\Gamma$ in hand, characterized by the sextuple $(0,0,12,13,14+23,24+15)$, which records the exterior derivatives of a basis of $\Gamma$-invariant ...
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1answer
61 views

$(0,1) \simeq [0,1] $ but not homeomorphic

I have to show that $(0,1) \simeq [0,1] $ but not homeomorphic, where $\simeq $ means homotopically equivalent. I have done the not homeomorphic part by showing that $(0,1)$ is not compact but ...
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1answer
26 views

Homotopy $I^2 \rightarrow S^1$ lifting lemma proof

In case of a homotopy $h: I^2 \rightarrow S^1$ we can define lifting as such an $\tilde{h}: I^2 \rightarrow \mathbb{R}$ that $e^{i\tilde{h}}=h$. The existence of $\tilde{h}$ requires a proof. A way to ...
3
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1answer
50 views

Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
0
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1answer
49 views

Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
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46 views

multiplication in H-space and loopspace of the H-space

Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy. Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $ \circ $ on ...
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45 views

Using transfinite induction to compute fundamental groups

I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen. Let's ...
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1answer
42 views

Are the space of paths with two given endpoints in a contractible space, contractible?

This question is inspired by an answer to Nitrogen's answer to my Are the path connected components of $\Omega S_1$ contractible? . Here we are asked whether the space of paths in $\mathbb{R}$ ...
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1answer
47 views

Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has ...
3
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1answer
38 views

Are the path connected components of $\Omega S_1$ contractible?

Let $\Omega S_1$ be the space of loops of $S_1$ based at $x_0 \in S_1$ with the topology of uniform convergence. We know that the path connected components of this space are in one to one ...
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73 views

Correct definition of model category

When answering this question, In a model category, is the full subcategory of fibrant objects a reflective subcategory? I realized that I wasn't even sure what the correct definition of a model ...
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1answer
32 views

Extend a map over a $n+1$-cell IFF $f_nϕ$ is nullhomotopic.

Prove: If the map $f_n$ is defined on the $n$-skeleton $X_n$ and you want to define it on an $n+1$-cell with attaching map $ϕ:S_n→X_n$, then you can do so if and only if $f_nϕ$ is nullhomotopic. I ...
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1answer
55 views

$T^2-D$ does not retract to the boundary $\partial D$

First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of ...
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33 views

Gluing along infinitely many trivial cofibrations

I am in a situation where I have a space Y obtained from a space X by gluing on infinitely many trivial cells. That is, I have a collection of maps $f_\alpha: A_\alpha \to X$ each of which is a ...
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13 views

Equicontinuous homotopies of families of uniformly equicontinuous functions

Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of ...
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20 views

Show that $h$ is homotopic to the identity map relative to $C$.

This is problem 5.3 and 5.4 in Armstrong's Basic toplogy. They are very much connected and i have solved problem 3. 3: Let $D$ be the disc bounded by $C$, i.e. $S^1$, parametrize $D$ using polar ...
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72 views

Why is the punctured plane not homotopic to the circle?

I know that the fundamental group of $X = \mathbb R^2 \setminus \{(0,0)\}$ is the same as the fundamental group of the circle $Y = S^1$, namely $\mathbb Z$. However, $X$ and $Y$ are not homotopic, ...
3
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2answers
64 views

Elementary geometric characterization of spheres?

I've read the following two theorems. Theorem. A compact connected metric space whose points are cuts points with the exception of at most two is homeomorphic to the unit interval. Theorem. A ...
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70 views

How much algebra one needs to study algebraic topology and homotopy theory?

I wonder how much algebra(group theory, abstract algebra, linear algebra, ring theory, field theory) is assumed as a prerequisite in most of the modern algebraic topology text. For example, these ...
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2answers
41 views

Trivial loop on the $1$-Skeleton

Following Hatcher's proof of Hurewicz Theorem (version of 1999) we arrive at the point that we must show that the loop in the picture, created following the path $0,1,2,3,1,0,1,3,0,3,2,0,2,1,0$, is ...
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0answers
16 views

Munkres positive linear map definition of path product (page 328)

I'm confused by Munkres' definition of the path product using the positive linear map. He defines the positive linear map $p: [a,b] \rightarrow [c,d]$ to be the unique map of the form $x \mapsto mx + ...
4
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1answer
55 views

Homology with local coefficients as a functor from pointed, path-connected spaces and $\pi_1$-modules.

A local system of coefficients on a space $X$ is a functor $F\colon \Pi(X)\rightarrow Ab$ from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups ...