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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3
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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 ...
0
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1answer
53 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 ...
1
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1answer
26 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, ...
2
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0answers
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 ...
2
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1answer
54 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 ...
2
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1answer
54 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, ...
4
votes
2answers
103 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 ...
0
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2answers
56 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 ...
3
votes
0answers
34 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.
3
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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 ...
2
votes
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, ...
8
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1answer
72 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
votes
0answers
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
votes
1answer
54 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 ...
0
votes
1answer
146 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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0answers
128 views

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
votes
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
votes
1answer
52 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 ...
2
votes
1answer
44 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 ...
7
votes
0answers
95 views

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. ...
1
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0answers
30 views

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 ...
0
votes
1answer
74 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 ...
0
votes
3answers
62 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 ...
2
votes
1answer
100 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 ...
1
vote
0answers
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 ...
4
votes
1answer
111 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 ...
10
votes
2answers
265 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 ...
0
votes
2answers
44 views

How to search the set of papers whose references contain a given preprint?

I am reading a preprint titled Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (available at Cohen's web page) Now I need to find all papers whose references contain this preprint. Is ...
0
votes
0answers
50 views

cohomology homomorphism between grassmannians induced by inclusion

Let $i:G_k(\mathbb{R}^n)\to G_k(\mathbb{R}^\infty)$ be inclusion of grassmannians. Then $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
3
votes
1answer
70 views

Does every elliptic cohomology theory represent a complex-orientable $E_\infty$-ring spectra and vice-versa?

The last paragraph in Two-Vector Bundles and Forms of Elliptic Cohomology remarks that neither the spectrum $K(ku)$ nor tmf is complex orientable. In the case of $K(ku)$: "...the unit map for $K(ku)$ ...
2
votes
0answers
38 views

pseudonatural vs natural

By a general result of steve lack's article, I know that there is a nice adjunction between the 2-category of 2-functors (and 2-natural transformations) and the 2-category of 2-functors and ...
1
vote
0answers
46 views

2-natural equivalences

This is probably a trivial question. Let D be a 2-category. Consider a model category X. When is it possible to consider a 2-categorical structure in X such that the equivalences of 2-Fun (D, X) are ...
0
votes
1answer
23 views

homology of suspension

Let $\Sigma$ be suspension. For any CW-complex, or topological space, does the reduced homology satisfy $$ \tilde H_*(\Sigma^k X)=s^k\tilde H_*(X)? $$ Here $s^k H$ is a copy of $H$ such that an ...
0
votes
0answers
22 views

Configuration space of product spaces

Let $M,N$ be manifolds. Let $F(M,n)$, $F(N,n)$ be ordered configuration spaces of order $n$. Let $F(M,n)/\Sigma_n$, $F(N,n)/\Sigma_n$ be the unordered configuration spaces of order $n$, for ...
0
votes
2answers
67 views

Mobius strip homotopically equivalent to filled out torus

Why is the Mobius strip homotopically equivalent to $S^1 \times D^2$? The latter is a filled out torus. How do we deform it to get the Mobius strip? Im not very familiar with homotopic equivalence, so ...
2
votes
1answer
58 views

Combining homotopies

I have two homotopies $H,G:D^n \times I \to Z$ with $H(x,0) = f(x)$, $H(x,1) = f'(x)$, $G(x,0) = f'(x)$, $G(x,1) = g(x)$ for some maps $f,f',g:D^n \to Z$. $G$ has the additionnal property of being ...
2
votes
0answers
55 views

Explicit expression for the topological invariant of O(n)

I learned that the fundamental group of $O(n)$ is $\Bbb{Z}/2\Bbb{Z}$ (for $n>2$). What is the explicit expression for its topological invariant? To be specific: Given a smooth path ...
3
votes
3answers
89 views

Homology and Homotopy in the Plane

Suppose we're living in the plane minus (possibly infinitely many) isolated points, which I'll call poles. Intuitively, the following two statements seem reasonable: Loops in the plane are homotopic ...
2
votes
1answer
40 views

cohomology monomorphism between grassmannians and product of projective spaces

Let $S^1\times\cdots \times S^1( n\text{ times })=\prod_n U(1)\to U(n)$ be the inclusion. This induces a map between classifying spaces $$ \prod_nBS^1\to BU(n).$$ i.e., $$ ...
3
votes
1answer
58 views

Postnikov towers for non-CW spaces

In the literature, Postnikov systems seem to be defined always in the setting of CW complexes. Looking at the proofs, it is not clear to me, why this assumption should be necessary. Question: Does ...
1
vote
1answer
68 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow ...
0
votes
0answers
28 views

fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
0
votes
1answer
33 views

Show that the cylinder is not ambient isotopic to the Mobius band.

Here is my definition for ambient isotopy: We say if there is an orientation preserving piecewise linear homeomorphism $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ (or replace $\mathbb{R}^3$ with $S^3$) ...
-1
votes
1answer
90 views

Homotopy Groups for Categories

With this observation in mind how far are we from defining $\forall \mathcal{C} \ \text{category}\ \pi_1(\mathcal{C})$? Let me be more clear. Let be $n$ the following category $0 \rightarrow 1 ...
2
votes
1answer
120 views

Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
1
vote
0answers
57 views

Homotopy equivalence of pushouts of topological spaces

Let $h \colon A \to B$ and $r \colon S^{n-1} \to A$ be continuous maps. Assume that $h$ is an homotopy equivalence, prove that $$ D^n \cup_{r} A \simeq D^n \cup_{h \circ r} B$$ where $D^n ...
0
votes
0answers
31 views

Are $z^n$ and $p(z)/|p(z)|$ homotopic?

Let $p:\mathbb C\longrightarrow \mathbb C$ be a complex polynomial with no zeros and degree $n$. Is it true that the maps $f, g:S^1\longrightarrow S^1$ given by $$f(z)=z^n\quad \textrm{and}\quad ...
1
vote
1answer
36 views

Free loop space of classifying space as a disjoint union of classifying spaces of centralizer proof reference request.

I am looking for a reference for the proof or explanation of why for a discrete group $G$ we have that the free loop space of its classifying space is the disjoint union of centralizeers of $g$ where ...
2
votes
1answer
41 views

Fundamental group of sphere

To show that $S^3$ is not diffeomorphic to $S^2 \times S^1$, I'd like to say that their fundamental groups are not the same. So $\pi_1(S^3)= 0 $ but why is $\pi_1(S^2 \times S^1) = Z $ ?
2
votes
1answer
104 views

Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities). the isomorphism classes of complex line bundles over $X$ the homotopy classes of ...