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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2
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0answers
13 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.
-1
votes
0answers
54 views

How to prove that an injection from a sphere into a Euclidean space is homotopic to a constant?

How to prove that the injection $i: S^{m-1}\rightarrow \mathbb{R}^m$ is homotopic to a constant ? Where $S^{m-1}=\{x\in \mathbb{R}^m, |x|=1\}$ Thank you.
2
votes
1answer
43 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
51 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
votes
1answer
50 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, ...
5
votes
0answers
57 views
+150

Action of $\operatorname{Aut}(G)$ on the Borel construction

I am interested into the (say real) regular representation $\rho$ of $G=(\mathbb{Z}/p)^n$. Considering the universal vector bundle $EG\rightarrow BG$, the Borel construction with the regular ...
4
votes
0answers
18 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 ...
2
votes
1answer
41 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
49 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 $X/A$ ...
4
votes
0answers
76 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
91 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
32 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
38 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
84 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
vote
0answers
26 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
65 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
50 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
59 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
37 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
72 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 ...
9
votes
2answers
187 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
0answers
20 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
28 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
23 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
25 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
41 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
18 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
17 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
0answers
23 views

maps between suspension of complex projective spaces and special unitary groups

How to do the following question? I get totally lost... this question is given by the professor in our final exam paper.
0
votes
2answers
46 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
47 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
51 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
69 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
33 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
48 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
35 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
20 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
0answers
12 views

Proof that $ X \times I$ is a CW complex

I am trying to prove that the mapping cylinder is a CW complex and to start, I need to show that $ X \times I$ is a CW complex, where $X$ is a CW complex . I haven't seem the proof that the product of ...
0
votes
1answer
27 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
83 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
84 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
35 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
28 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
25 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
28 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
65 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 ...
3
votes
1answer
26 views

How to show that homotopy of chain maps respects composition?

Given the homotopic pairs of chain maps $f_1 \simeq f_2 : A_* \to B_*$ and $g_1 \simeq g_2 : B_* \to C_*$, show that $g_1 \circ f_1 \simeq g_2 \circ f_2: A_* \to C_*$. $f_1 \simeq f_2$ means that ...
1
vote
1answer
65 views

On chain homotopy equivalence

I just learnt the notion of chain map and have the following question. Let $C=(C_n,\partial_n^C)$ and $D=(D_n,\partial_n^D)$ be chain complexes of abelian groups with boundary maps $\partial_n^C$ and ...
2
votes
1answer
133 views

Question about homotopy equivalence

I have this proof but I don't understand why $i\circ j$ induces a homotopy equivalence, and how to see $j_*$ is injective at the level of homology? $X$ is a Banach space
0
votes
1answer
42 views

Proving that induced homomorphism is an isomorphism

Let $A \subset X$ and let $j: A \to X$ be the inclusion map. Let $f: X \to A$ be a continuous map. We suppose there is a homotopy between $j \circ f$ and the identity map on $X$. We are to show that ...