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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cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
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Fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
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Does the 2-functor $PsAlg\to \mathfrak{X} $ reflect equivalences?

Consider a $2$-monad $ T: \mathfrak{X}\to \mathfrak{X} $ and consider its 2-category of pseudoalgebras $PsAlg$. There is a forgetful functor $ U: PsAlg\to \mathfrak{X} $. Does this forgetful functor ...
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64 views

Hopf invariant and homotopy groups of spheres

I am trying to understand how to use Hopf invariant, to calculate $\pi_{4n-1}(S^{2n})$. I've started with defining a new space $X$ adjoining $D^{4n}$ to $S^{2n}$ via a map $\phi\in\pi_{4n-1}(S^{2n}) ...
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Sequence of cofibrations gives strict commutativity of given triangle

The problem is the following: Suppose $X_0\rightarrow X_1\rightarrow\cdots$ is a sequence of cofibrations. We will denote $f_n:X_n\rightarrow X_{n+1}$. Assume that we have also maps ...
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40 views

Null-homotopic maps

Assume that $[\alpha]\in\pi_n(X,x_0)$. I want to prove the following: $[\alpha]=0$ if and only if $\alpha:S^n\rightarrow X$ extends to a map $D^n\rightarrow X$. Can someone help me with this proof? ...
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56 views

Homotopy Type of Surface of Genus g

Need help with the following exercise; "Let M be a compact orientable surface of genus g. Prove that M with a point removed has the same homotopy type as 2g circles with a point in common." I have ...
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65 views

Relative $CW$-complexes and Serre fibrations

Suppose we have a (continuous) map $p:E\rightarrow X$. Assume that $p$ has the right lifting property with respect to any relative $CW$-complex $i:A\rightarrow B$. I want to show that $p$ is a Serre ...
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83 views

An example of why this homomorphism of homotopy groups is not injective.

For a given path-connected space $X$, I recently learned that one could construct a CW complex $X_{1}$ by considering each generator $j: S^{q} \rightarrow X$ of $\pi_{q}(X)$ and setting ...
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44 views

Reference for secondary cohomology operations

I am learning some homotopy theory and am currently reading Mosher and Tangora. I love the content of this book, it's very terse and comes straight to the point. At the same time I find it very ...
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1answer
115 views

Is this morphism of spectra zero in the stable homotopy category?

Let $f\colon A\to B$ a morphism of spectra and suppose that both spectra $A$ and $B$ have only one non-zero stable homotopy group $\pi_n$, more precisely $$ \pi_k(A)=0=\pi_k(B) $$ for $k\neq n$. ...
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40 views

null homotopy of 1-skeleton of simply connected simplicial complex induced by homotopy identity

I am reading a book suggesting that the following is true: Let $X$ be a simply connected (maybe finite) simplicial complex. Then there exists a continous map $$g : X \to X,$$ homotopic to the ...
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24 views

Basic question about abelianization of Homotopy Groups and Homology [duplicate]

When precisely, is the homology group: $$H_n(T)$$ of a topological space, $T$, isomorphic to the abelianiation of the corresponding homotopy group $\pi_n$? Does this only occur when $n=1$, or is ...
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74 views

mapping cone and cylinder

Given a map of spaces $f:X \to Y$, the mapping cylinder is the adjunction space $$cyl(f)=(X \times [0,1]) \cup_f Y$$ where we regard $f$ as a map $f: X \times \{1\} \to Y$.\ On the other hand the ...
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57 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
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1answer
43 views

Why a convergent succesion does not have the same homotopy type of a CW-Complex?

The question is pretty much in the title; If my space is $\{1/n\}_{n\in \mathbb{N}} \cup \{0\} $ why it isn't homotopically equivalent to a CW-Complex?
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63 views

Show that $\mathbb{R} P^3$ is not homotopy equivalent to $\mathbb{R} P^2 \vee S^3$.

I'm studying for an oral qualifying exam in algebraic topology, going through questions in various tests published on the interwebs. Here's a rather straightforward question from this exam that is ...
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58 views

Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
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1answer
59 views

Smash product and tensor product of groups

The smash product acts like a 'tensor product' in the category of pointed spaces (i.e. when the spaces are locally compact Hausdorff smashing is associative and satisfies a tensor-hom adjunction). ...
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1answer
66 views

Attaching cells gives isomorphism of homotopy groups

I want to prove the following statement: Let $(X, x_0)$ be a pointed space, and let $X' = X\cup_{\alpha} e^{n+1}$ be obtained from $X$ by adjoining an $(n + 1)$-cell. Then the inclusion $i : ...
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24 views

Dual of path object

For a topological space X, what might be the dual of the path space $X^I$ of X? Does it make any sense to think of the topological cylinder X x I over X as dual to the path space over X?
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2answers
43 views

Fibration $p_1:P(Y,y_0)\to Y$ has section iff $Y$ is contractible.

Let $P(Y,y_0) = \{ \omega : \omega(0) = y_0 \}$ be path space let's consider a fibration $p_1:P(Y,y_0)\to Y$ such that $\omega \mapsto \omega(1)$. Show that there exists $s: Y \to P(Y,y_0)$ such ...
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1answer
50 views

Killing homotopy groups

I am basic with homotopy theory and especially with CW-approximation, Postnikov and Whitehead towers. In the proof of such things one need the following result on and on: Let $X'$ be obtained from ...
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42 views

Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?

In a model category, when weak equivalences are inverted, nothing else gets inverted. It follows that weak equivalences satsify 2-out-of-6. But the first sentence takes some work to show. Is there a ...
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1answer
52 views

Homeomorphism “induced” by Lifting map. [closed]

I was going over this post :Explicit form of a lift $\tilde f: \tilde X_1 \to \tilde X_2$ of a continuous map $f: X_1 \to X_2$ $$\require{AMScd}\begin{CD} \tilde X_1 @>\tilde f >> \tilde ...
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1answer
22 views

Discrete simplicial spaces are fibrant

As the title suggests, I would like to understand why should a discrete simplicial space be fibrant. Let me be more precise. Consider the category $\textbf{sSet}^{\Delta^{op}}$ of simplicial spaces, ...
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26 views

CW approximation of $n$-connected space

I want to prove the following lemma: Let $X$ be a n-connected space. Then there exists a CW-approximation $f:K\rightarrow X$ such that $K$ has trivial n-skeleton. What I have done so far: By ...
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1answer
53 views

Explicit form of a lift $\tilde f: \tilde X_1 \to \tilde X_2$ of a continuous map $f: X_1 \to X_2$

This is embarrassingly simple for most, but I am a High School student trying to teach myself, and I am having trouble figuring it out: In the post Basic question about lifting maps to covering ...
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1answer
53 views

Homotopy class of maps to a complex projective space

Let $M$ be a closed oriented smooth 4-manifold. Denote by $[M, \mathbb{C}P^{\infty}]$ homotopy classes of continuous maps from $M$ to $\mathbb{C}P^{\infty}$. I would like to know how to show $$ [M, ...
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93 views

Fundamental groupoid

Let $(X,x_0)$ be a pointed topological space. The homotopy groups $\pi_n(X,x_0)=Hom((S^n,s_0),(X,x_0))$ are groups because $S^n$ is a cogroup object in the pointed homotopy category. Removing the ...
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55 views

What does “modulo a homotopy” mean?

From what I understand, the fundamental group of a topological space $X$ with base point $x_0$ is the set of all equivalence classes of continuous paths in $X$ that start and end at $x_0$. Formally ...
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31 views

The images of two non homotopic to identity maps intersect

How could one prove that images two maps $f,g:\mathbb RP^4 \to \mathbb RP^7$ which are not homotopic to trivial map have nonempty intersection.
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1answer
76 views

How do modern categories of spectra manage to avoid “cells now, maps later”?

In Adams' definition, a map $f: X \to Y$ of CW spectra consists of a cofinal subcomplex $X'\subseteq X$ and maps $f_n: X'_n \to Y_n$ that commute with the structure maps. This definition is reproduced ...
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0answers
67 views

Dual of path in a space.

Is there a notion dual to the notion of a path in a topological space? Given that a path in a space $X$ is a continuous function from the interval $[0, 1]$ to X, what would the dual of this notion be, ...
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1answer
71 views

Is this contour continuously deformable into a circle?

As an exam question, we had to solve the integral of $\frac{1}{z}$ over the following contour: (The contour is a sequence of straights arcs joining -1, -$\frac{i}{2}$, $\frac{1}{2}$, i, ...
4
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23 views

Viewing Homotopies as Paths in $\mathcal{C}^0(X,Y)$

When I think intuitively about homotopies, I think about them as paths between two functions. This is more comfortable and suggestive than any categorical talk about "morphisms between morphisms", so ...
3
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1answer
53 views

Understanding the Homotopy Invariance of Fiber Bundle

I'm trying to understand the proof of Theorem 2.1 in "The Topology of Fiber Bundles" found online at http://math.stanford.edu/~ralph/fiber.pdf. What I don't understand is how do we actually define ...
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1answer
140 views

How to prove the Cone is contractible?

Let $\sim$ be an equivalence relation on a topological space $X$ with a subset $A ⊂ X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then define $X/A$ ...
6
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Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective embeddings $f:K_1\rightarrow K_2, g:K_2\rightarrow ...
3
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0answers
105 views

Maximum diameter for the preimage of a point when the degree is not 1 or -1

When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the ...
2
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1answer
50 views

Equivalence of unoriented knots by ambient isotopy

I'm trying to understand the equivalence of unoriented knots in oriented 3-manifolds for my thesis, and getting confused. I have not found a satisfactory definition of this equivalence. My ...
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1answer
42 views

Homology as Boundary of “Submanifold”

In the plane, imagine a horizonal figure eight, $\infty$. Let $\alpha$ be the curve which is convex from the leftmost point of the figure to its "middle", and concave from the middle until the ...
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History of the term “anodyne” in homotopy theory

There is a notion of an anodyne morphism (usually of simplicial sets). This type of morphisms is used, for instance, to establish basic properties of quasicategories. But the name is quite mysterious. ...
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Homology and Homotopy in the Plane II

This question arose from Homology and Homotopy in the Plane, where it was one of several questions asked (but not answered). I'm posting it separately so I could accept one of the answers there. Is ...
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1answer
70 views

what is a path that cover all of $S^n$?

Here is the meaning of "cover" which I can't understand: Prove that if $n\ge 2$, then $S^n$ is simply connected. hint: Use Exercise 2.5 to show that every loop in S" is homotopic to a loop that does ...
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3answers
61 views

Geometric Homotopy as Chain Homotopy

In Can we think of a chain homotopy as a homotopy, I learned that chain homotopy can be defined in an analogous fashion to homotopy, i.e from the product with an interval object etc. What about the ...
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1answer
92 views

Converse to the Eilenberg-Steenrod theorem?

For the purposes of this question, a homology theory is a covariant functor from the homotopy category of finite pointed CW complexes to graded abelian groups, and a collection of connecting ...
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55 views

If $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$

I want to show that if $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$. It feels like I'm missing something really obvious. $X$ is homotopic to $Y$, so there exists $f: X \to Y$ and $g: Y ...
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1answer
104 views

Reference Request: Thom Spectrum of a virtual vector bundle

Given a vector bundle $\xi$, one can construct a Thom spectrum $\mathbb{T}h(-\xi)$ associated to its additive inverse. I unsuccessfully browsed through literature to find a reference for an explicit ...
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2answers
255 views

What is the best path to learn Category theory and Type thoery?

I have little background in Programming in functional languages and wanted to learn type theory. I started with taking Homotopy type theory class Online videos of Robert Harper. I thought rather ...