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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Basic localizers contain adjoint functors

I'm struggling with a property of basic localizers, namely that adjoint functors are weak equivalences. Recall that a basic localizer $\mathcal W$ is a subset/subclass of the arrows of the category ...
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45 views

Mapping $\mathbb R^n - \{0\}$ to $S^{n-1}$

How might one map $\mathbb R^n - \{0\}$ to $S^{n-1}$ ? I found this in a primer on homology where it is proved that the to spaces are homotopy equivalent, as an example of removing a single point ...
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Surjectivity and exactness on higher homotopy groups

If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$. If $f\colon X \rightarrow Y$ is ...
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61 views

A map $f: X\rightarrow Y$ is a homotopy equivalence if and only if $h\circ f,f\circ k$ are homotopy equivalences of $X,Y$ respectively.

Show that $f\colon X \rightarrow Y$ is a homotopy equivalence if and only if there exist maps $k,h\colon Y\rightarrow X$ such that $f\circ k$ is a homotopy equivalence of $Y$ to itself, and $h\circ f$ ...
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Combinatorial definition of the homotopy groups of a quasi category?

The homotopy groups of a Kan complex are defined combinatorially, and they coincide with the homotopy groups of the geometric realization. For a general complex $X$, homotopy isn't an equivalence ...
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$[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets

how to show: if $F \to E \to B $ is a fibration then for any space $X$ the sequence $[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets. any hints, thanx.
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$\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$

I need help with this problem Let $\sum X =C^+X \cup_X C^-X$ be the union of two cones on $X$ with a common base. Show that $\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$. Dose this give an ...
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Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
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59 views

Homotopy on the unit circle

I am trying understand why the identity function on the unit circle $X=\{(x,y): x^2+y^2=1\}$ is not homotopic to $f: X \to X$ where $f(z)=(1,0)$ for all $z\in X$.
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problems with proving that f and g are homotopic.

i need to give an example of 2 continuous functions $f,g: X \rightarrow Y$ which are not homotopic, with: $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$ and i need to show how many homotopical ...
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50 views

Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
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28 views

Showing two things are homotopic to each other

I want to show that $\mathbb{C} - \{0\} \simeq S^1$ and the unit square is $\simeq S^1$ where $\simeq$ is homotopic in this case. In other words I want to find an equation for each sort of speak that ...
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1answer
31 views

Spaces homotopy equivalent to $A_{\infty}$-spaces

I ask this question after reading Peter May's "Geometry of Iterated Loop Spaces", where the problem is definitely hinted at but I couldn't find a definite answer. Recall a symmetric operad ...
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42 views

Homotopy type vs. weak homotopy type, and repercussions for EG

This is another basic question. I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point ...
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20 views

Geometric realization of function complexes of simplicial sets

Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called ...
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1answer
54 views

Fundamental Group equaling 0

Let $X$ be a space for which $\pi(X,x)=0$. If $f,g$ are two paths in $X$ with $f(0)=g(0)=x$ and $f(1)=g(1)$, why is $f$ equivalent to $g$?
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Proving homotopy of paths

Let $f$ be a path in $X$ and $h:[0,1] \mapsto [0,1]$ a continuous mapping with $h(0)=0$ and $h(1)=1$. How can I prove that $f$ and $fh$ are homotopic relative to the endpoints?
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42 views

A certain homotopy equivalence…

A few friends and I have been stuck on this old qualifying question for quite some time now... Let $D$ be the diagonal subspace of $\Bbb S^2 \times \Bbb S^2$. Show that the projection onto the first ...
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47 views

Why do we ask that condition in Kan complex?

Let $\{X_n\}_{n=0}^\infty$ be simplicial set with faces $d_i:X_n\to X_{n-1} $, a simplicial set is called a Kan Complex if for any $x_0,...,x_{k-1},x_{k+1},...,x_{n+1}$ if $d_i(x_j)=d_{j-1}(x_i)$ for ...
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38 views

Homotopy equivalences

If $x\in X$, let $C(x)$ the path component of $x$ (the biggest path connected set containing $x$), and similarly if $y\in Y$. Let $C(X)$ and $C(Y)$ the family of all path components of $X$ and $Y$. ...
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Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals?

Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals ? So by dual, I mean the linear maps on $H_{k}$. I need this to understand the Poincare duality i.e. $H_{k}\cong ...
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Proving the existence of some deformation retract

I am trying to find out if the set $ S ^n \times S^n \setminus \left\lbrace (x,x) \mid x \in S^n \right\rbrace$ deformation retracts onto the subspace $\left\lbrace (x,-x) \mid x \in S^n ...
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78 views

Solving an exercise in Milnor-stasheff's “characteristic classes”

I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let $ M =S^n $ be the $n$-sphere and let $TM$ be its tangent ...
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43 views

Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
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simplicial commutive rings as algebra objects in a symmetric monoidal category

Do simplicial commutative rings arise as the commutative algebra objects in some symmetric monoidal infinity category? If I understand correctly, I think $E_\infty$ rings are the commutative algebra ...
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21 views

homotopy - maintaining curvature signs

I have the following dilemma! Say, $f_1=\sqrt{1-x^2}$, and $f_2=-\sqrt{1-x^2}$ are two continuous functions on $[-1,1]$ Lets define another function by $F = tf_1 + (1-t)f_2$ where $t=[0,1]$ ...
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57 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
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47 views

Does $\pi_0$ preserve infinite products?

The functor $\pi_0\colon \operatorname{sSets}\to \operatorname{Sets}$ has value on a simplicial set $X$ the coequalizer of the two boundary morphisms $d_0,d_1\colon X_1\to X_0$. It is easy to see ...
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28 views

“Homotopy theory” on finite topological spaces?

My question concerns finite sets carrying a not-necessarily-discrete topology. I'm wondering if there's an analogue for homotopy theory where the role of $S^n$ is played by some other, finite set. (My ...
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31 views

Homotopy between two homomorphisms and homology

If I have two chain complexes $C$ and $D$ and I suppose that there is a homotopy between $\phi, \psi:C \rightarrow D$ (i.e there is a sequence of homomorphisms $(K_n: C_n\rightarrow D_{n+1})$ such ...
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35 views

existence of a cofiber sequence

can anyone help me with this problem. thanx. Show that there are cofiber sequence $S^{n+3} \to S^{n+2} \to \sum^{n}\mathbb{C}P^2$ for each $n \in \mathbb{Z}^+$. Conclude that a space of the form ...
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Explanation for a line from a MathOverflow answer

Sometimes I see questions answered on MathOverflow in such a way that I don't really understand the answers. Sometimes I work out what they mean, and other times I can't. I'd like to ask for more ...
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1answer
23 views

Eilenberg-Mac Lane and classifying spaces

What can we say about An Eilenberg-Mac Lane space $K(G,n)$ is a classifying space $BG$. When it could be true? For what kind of $G$? For what values of $n$? References are welcomed.
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2answers
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homotopy between two functions

Let us discuss this problem: Let $A=\{a_{1},a_{2},\ldots,a_{n}\}$, $B=\{b_{1},b_{2},\ldots,b_{n}\}$ and $C=\{c_{1},c_{2},\ldots,c_{n-1}\}$ be discrete finite sets embedded in a unit sphere ...
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The two-sided bar construction on simplicial model categories

I am trying to understand example 4.5.7 here http://www.math.harvard.edu/~eriehl/cathtpy.pdf . More precisely, I am curious: how is the maps defined by composition and why is it a section of $d_0$? ...
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Why is $\pi_1(X,x_0)$ a group?

I want to show that $\pi_1(X,x_0)$ is a group. I am told that $e(t) := x_0$ is the identity element. Now, I am struggling to show that it is an identity element, and also that the inverse of an ...
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41 views

Path-homotopic definition.

Given two paths $f,g: [0,1] \mapsto X $ Then their product is defined by $f\cdot g := \begin{cases} f(2t) , \space 0\leq t \leq \frac{1}{2}\\ g(2t-1) ,\space \frac{1}{2} \leq t \leq 1\\ \end{cases} ...
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Is there any standard terminology for the quotient of a topological group by the connected component of the identity?

If $G$ is any topological group, then the connected component of its identity is a closed normal subgroup $H$. It follows that $G/H$ is a totally disconnected topological group. Often, $G$ will be ...
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1answer
43 views

Example for a space that is contractible to precisely one of its points

Give an example for a space that is contractible to one of its points and is not contractible to another of its points. I am really curious about that space, I have thought about tree with $n$ ...
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1answer
38 views

Why is a spectrum $X$ with only $\pi_0X\neq 0$ equivalent to an Eilenberg-MacLane spectrum?

For an abelian group $G$, the Eilenberg-MacLane spaces $K(G,k)$ assemble to a spectrum $HG$ with $HG_k=K(G,k)$. This spectrum has the property that $\pi_0 HG=G$ and $\pi_{n}HG=0$ for $n\neq 0$ where ...
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Lowest norm solution to a system of polynomial equations

I have a system of cubic equations: $$0=A_0+A_1 x+A_2 ( x \otimes x ) + A_3( x \otimes x \otimes x )$$ where $\dim A_0 = \dim x$ (so there are as many equations as unknowns). You may assume that the ...
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What is the difference between multiplication and direct sum on homotopy groups of spheres?

In Allen Hatcher's book Algebraic topology he states that factoring out 2-torsion $\pi_{i}(S^{2n})\cong\pi_{i-1}(S^{2n-1})\times\pi_{i}(S^{4n-1})\:\forall n$ but in his book Spectral Sequences in ...
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56 views

Quasi circle is not contractible

I'm trying to show that the quasi circle (picture below) doesn't have the homotopy type of a CW complex. I proved that all homotopy groups are zero. Now I need to show that it is not contractible to ...
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134 views

how should I show that it is wedge of infinite circles?

I know that the shape that we see it below is homotopy equivalent of wedge of infinite circles,so the fundamental group of it is $\prod _{1}(\vee _{\alpha \in A}S^{1})=\ast _{\alpha \in A ...
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75 views

Spaces with different homotopy type

I want to show that the spaces $ S^1 \vee S^1 \vee S^2$ and $S^1 \times S^1 $ do not have the same homotopy type. I calculated their homologies and cohomologies and they turn out to be equal. So I ...
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49 views

Question on the uniqueness of a homotopy colimit up to unique isomorphism

Let me first give an abstract definition of the homotopy colimit. Let $C$ be a cofibrantly generated model category and let $D$ be a small category. There is an adjunction $$ ...
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30 views

Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
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14 views

Terminology for homotopies which stay inside some finite stage of a union

Sometimes it happens that you have a sequence of topological spaces each contained in the next $$ X_1 \subset X_2 \subset X_3 \subset \ldots$$ and you want to talk about things like homotopy in the ...
3
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1answer
87 views

Fibrations induced by deformation retractions

Given a map $f: A \to B$ and a homotopy equivalence $g: C \to A$, I wish to show that $E_f \to B$ and $E_{fg} \to B$ are fiber homotopy equivalent, where, given a function $f: A \to B$, $E_f$ is the ...
3
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30 views

Derived pseudo-functor

Let $ \mathfrak {X}\to \mathfrak{Y} $ be a pseudofunctor (in which $\mathfrak{X} $ is a model category and $\mathfrak{Y} $ is a bicategory). I would like to understand when there is a derived functor ...