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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1answer
52 views

Why is the unit circle, $\mathbf{S}^1$, a deformation retract of $\mathbf{R}^2$ minus any point?

It is clear that $\mathbf{S}^1$ is a deformation retract of $\mathbf{R}^2\setminus\{0\}$ since we can consider the straight line deformation retract $H\colon (\mathbf{R}^2\setminus\{0\}) \times [0, 1] ...
0
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28 views

inclusion of homotopy fiber and induced map on homology group

Given a fibration $F \to E \to B$, under what circumstances does the inclusion of the homotopy fiber into $E$, $F \to E$, induce injections on homology? The specific case I'm dealing with involves the ...
1
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0answers
17 views

Lifting a homotopy class $S^k\to X$ into a simplicial set $X$ which is not fibrant but satisfies some weaker horn filling condition

Let $X$ be a connected simplicial set. If $X$ is an Kan complex and $k\geq 0$, then every element $$ \tilde f\in\operatorname{Hom}_{Ho(sSet)}(S^k,X) $$ of the homotopy classes from $S^k$ to $X$ lifts ...
1
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1answer
27 views

Intersection preserves homotopy equivalence

Let $Z$ be a topological space with subspaces $X$, $Y$, $X'$ and $Y'$. Suppose that $X$ is homotopy equivalent to $X'$ and $Y$ is homotopy equivalent to $Y'$ do we have that $X\cap Y$ is homotopy ...
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39 views

Nth Homotopy Group Isomorphic to [T^n, X]

Following Spanier's book on algebraic topology chapter 1, section 6 about suspensions, I'm wondering about the following questions: 1) We know that $S^n$ is an H co-group for all $n\geq1$ because ...
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0answers
50 views

Homotopic family of curves

I stumbled over the following question. Imagine we have a two homotopic curves on the sphere $\mathbb{S}^1$ namely $\gamma_1,\gamma_2$. Then we can write them as $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
1
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1answer
69 views

Action of $S^1$ on homotopy groups of an $S^1$-space

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed ...
2
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1answer
51 views

Question about maps to $K(G,1)$

I have an unfortunately basic confusion about two results in Hatcher's Algebraic Topology. Let $G$ be an abelian group. Theorem 4.57 specialized to $n=1$ says that there is a bijection $$\langle X, ...
1
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1answer
18 views

How to check F:AxI->B is continuous

A and B are topological spaces.Let f and f' are continuous maps from A to B and homotopic.Then we need F:AxI->B,continuous,where F(s,0)=f(s) and F(s,1)=f'(s). Now my question is if we want to ...
1
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2answers
36 views

Convexity of space of integrals of homotopic paths

Let $B$ be a Banach space. Consider a set $Q \subset B$, a path $\gamma_0:[0,1] \to Q$ and the set of paths $\Gamma_0$ of paths $[0,1] \to Q$ homotopic (with ends fixed) to the path $\gamma_0$. Let ...
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0answers
46 views

How is the delooping of a groupoid constructed explicitly?

Let $G$ be a group, nlab's "delooping" page says that $G$ can be considered as a discrete groupoid in the $(\infty,1)$-topos $\infty$Grpd of $\infty$-grupoids, the delooping of $BG$ is then the ...
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17 views

show that exists exactly one n such that $\gamma_1$ is homotopic to $\gamma_2 = e^{2\pi int}$

Let $\gamma_1 $ be a closed path in $\Bbb{C}\setminus0$ such that $0 \in int (\gamma_1)$ show that exists exactly one $n \in \Bbb{Z}\setminus 0$ such that $\gamma_1$ is homotopic to $\gamma_2 : ...
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0answers
5 views

Homothetic Transformation for a Jordan closed set

The set $T$ is a closed Jordan set, and there exists a limit $$ T^{\infty}=\mathbf{t}_{0}+ \lim_{a \rightarrow \infty } h_{a}(T-\mathbf{t}_{0}) $$ of non-zero measure, where $h_{a}$ is a homothety ...
1
vote
1answer
20 views

Is the homotopy category of modules over a quasi-Frobenius ring (pre)additive?

Let $R$ be a quasi-Frobenius ring and $Mod_R$ the category of $R$-modules. One can prove that it admits a model structure whose weak-equivalences are stable equivalences, whose cofibrations are monos ...
3
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27 views

If weak-equivalences in a model category are generated by an equivalence relation $\sim$ on hom-sets…

Suppose that $C$ is a bicomplete category, and $\sim$ is an equivalence relation defined on every hom-set of $C$, compatible with composition. Then we can define the quotient category $C/\sim$ whose ...
3
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0answers
34 views

Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N\to M$ between closed oriented connected manifolds. Let $X$ and $Y$ be diffeomorphic submanifolds of $M$, and assume $h$ to be ...
2
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0answers
65 views

Is the Quillen Injective Model Structure on the category of positive cochain complexes of R-modules (co)fibrantly generated?

Denoted with $Ch^+_R$ the category of positive cochain complexes of R-modules (for a commutative ring $R$), it admits a model structure where: weak-equivalences are quasi-isomorphisms; cofibrations ...
3
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0answers
20 views

Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
0
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1answer
34 views

Hypothesis in homotopy equivalence inducing isomorphism in the fundamental groups

Let $X$ and $Y$ be topological spaces. If $f\colon X\to Y$ is a homeomorphism, then it induces an isomorphism $f_\sharp\colon\pi_1(X,x_0)\to \pi_1(Y,f(x_0))$. All good. As far as I know, the result ...
2
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0answers
39 views

Topological Boundary Map

In May, Concise Algebraic Topology, p. 108-109, for a cofibration $A \rightarrow X$ a "topological boundary map" is defined as the composite: $X/A \xrightarrow{\psi^{-1}} Ci \xrightarrow{\pi} \Sigma ...
1
vote
1answer
39 views

Is the reflective localization $L_WC$ of a category $C$ equivalent to $C$? What am I missing?

This is probably a dumb question but this is going over my head at the moment, I came here from nlab's entry on localization (http://ncatlab.org/nlab/show/localization). Let $C$ be a category, let $W ...
7
votes
1answer
98 views

Composition of homotopy classes with self-maps of spheres

Are there some general rules/formulas on the relation between the homotopy class $[f]\in \pi_i(S^n)$ and the homotopy class of the composition $S^i\stackrel{a}{\to} ...
3
votes
1answer
57 views

Choosing projective replacement to be functorial

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf ...
5
votes
1answer
71 views

Proving that the tangent vector of a simple closed curve rotates by $ 2 \pi$

I am trying to prove that if $\gamma(t)=(x(t),y(t))$ ,a function from the closed interval $[0,1]$ to $\mathbb{R^2}$ is a simple closed unit speed curve such that $\gamma '(0)=\gamma '(1)$. Then the ...
2
votes
1answer
40 views

$S^n$ is not a retract of the disk $D^{n+1}$ and Brouwer's Fixed Point Theorem.

I was trying to understand Hirsch's proof of this fact: "There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by ...
2
votes
1answer
45 views

Does symplectic K-theory $KSp$ have products?

The real and unitary topological $K$-theories are cohomology theories defined by the $\Omega$-spectra $KO$ and $K$ respectively. These are multiplicative theories with products deriving from the ...
2
votes
2answers
54 views

Another clarification about Thom-Pontrjagin construction

This is the second part of the following solved question. [I'm following Bredon's Book]. After explaining the idea behind the "desired" bijection we want to build, Bredon start dealing with the ...
2
votes
1answer
76 views

If the product of two homotopy equivalences is a homotopy equivalence are the factors homotopy equivalences?

The question says it all: Given two maps $f\colon A\rightarrow B$ and $f'\colon A'\rightarrow B'$, such that their product $$f\times f'\colon A\times A'\rightarrow B\times B'$$ is a homotopy ...
2
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0answers
18 views

How stable is the top cell of Lie group?

It is well know fundamental class of a compact lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable equivalence ...
9
votes
1answer
131 views

Gap between “fibration” and “fiber bundle”.

There are fibrations $E \rightarrow B$ which are not fiber bundles. Example: $E = [0,1]^2 / \text{middle vertical line segment}$ and $B=[0,1]$. In this example, $E$ has the homotopy type of a total ...
3
votes
1answer
67 views

Clarification about the Thom-Pontrjagin construction as explained in Bredon's book

In Bredon's book, at page 118-119, there is a little chapter about the Thom-Pontrjagin construction, and I'm trying to follow the reason depicted there. He starts with a map $f \colon R^{n+k}\to ...
4
votes
1answer
64 views

Showing that $\mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.

Intuitively, I can see why this is. I've found a few threads about this, but they only provide, for example, a deformation retraction of $I \times I$ to its diagonal $D = \{ (x,x) \in I \times I \}$, ...
2
votes
0answers
36 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
5
votes
1answer
85 views

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$. I need to show that a homotopy equivalence between them doesn't exist, but it seems like the homology groups of the spaces ...
4
votes
1answer
47 views

isotopy equivalence between manifolds

The definition below is from Encyclopaedia of Mathematics: Volume 6. Question: For any $n\geq 1$, is the $n$-dimensional closed cube $$[0,1]^n=[0,1]\times [0,1]\times\cdots \times[0,1]$$ isotopy ...
4
votes
0answers
55 views

Composition, fibrations?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. Suppose that $f$ is a homotopy equivalence. My question is, does it follow that $f$ is a fiber ...
1
vote
1answer
40 views

Deleting a contractible subspace is the same as deleting a point

Let $X$ be a topological space and $A$ and $B$ are subspaces of $X$. Suppose that $A$ is contractible. I know that taking the quotient does not affect the homotopy type, that is $X/A$ is homotopy ...
0
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0answers
35 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
2
votes
0answers
132 views

Deformation retract of a triangle

Let $X \subset \mathbb{R}^2$ be a triangle equipped with the topology induced by the euclidean topology on $\mathbb{R}^2$ and let $Y \subset X$ be the subset made of two sides of the triangle. I need ...
0
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0answers
50 views

$\Bbb R^5$ without two circles and line

How to prove that $\Bbb R^5$ without two $S^1$ and one $\Bbb R^1$ is homotopiclly equivalent to wedge of $(\Bbb R^5\backslash \Bbb R^1)$ and two copies of $(\Bbb R^5\backslash S^1)$? The only idea ...
3
votes
0answers
51 views

Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?). What ...
1
vote
2answers
49 views

Pullbacks and homotopy equivalences

Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it ...
0
votes
0answers
27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
1
vote
2answers
54 views

Action of the fundamental group

Suppose that $M$ is a smooth manifold. Is it true that the fundamental group $\pi_1(M)$ always acts on $M$? If so, how this action is defined? EDIT: Of course I want my action to be nontrivial, say ...
2
votes
1answer
41 views

Higher homotopy groups of wedge of circles.

Using van-kampen theorem, Fundamental group of wedge of n-circles is free group on n-generator. But I don't know how to calculate higher homotopy groups of wedge of spaces, in particular circles. I ...
4
votes
1answer
51 views

Fiber bundle with null-homotopic fiber inclusion

It is an exercise from Hatcher (exercise 31, page 392): For a fiber bundle $F \to E \xrightarrow{p} B$ such that the inclusion $F \hookrightarrow E$ is homotopic to a constant map, show that the long ...
4
votes
1answer
91 views

Fundamental group of open subsets of $\mathbb{R}^n$

Suppose that $U$ is an open subset of $\mathbb{R}^n$. What can be said about its fundamental group? I'm sure that the answer should be well known, since this is rather natural question.
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34 views

Show that there exist no retraction from $RP^n$ to $RP^k$ if n>k.

I am trying this problem from Hatcher's algebraic topology book(4.2.1). If r:X$\rightarrow$A is retraction then I know that this induces injective map in the fundamental group level through inclusion ...
2
votes
1answer
52 views

Zero in the Grothedieck group of the derived category

I have a problem. I was wondering whether there is a precise answer to the following question. Let $\mathcal{A}$ be an abelian category and $\mathcal{D}^b(\mathcal{A})$ its bounded derived category. ...
0
votes
0answers
25 views

how to calculate relative homotopy groups?

I am studying nth relative homotopy groups from Hather.For a pair (X,A) where A$\subset$X nth-relative homotopy groups is defined by homotopy class of maps$(I^n,\delta I^n,J^{n-1})$ ...