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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26 views

Is every infinite loop space a ring prespectrum?

Suppose you are given an infinite loop space $T_0$. The $i$th deloopings, $T_i$ then form a prespectrum with evaluation maps maps $\Sigma T_i \to T_{i+1}$. In the two examples that I know of, the ...
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27 views

Show that a set of homotopy classes has a single element

This is from Munkres section 51 problem 2b Given spaces $X$ and $Y$ let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. Show that if $Y$ is path connected, the set $[I,Y]$ has ...
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3answers
130 views

Homotopy equivalence of a space with the sphere

I have some trouble with the following problem. A space $X$ is obtained by gluing two $2$-cells to a circle $S^1$ using maps winding $2$-times and $3$-times around $S^1$. Show that $X$ is homotopy ...
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5answers
98 views

Examples of Same fundamental group but not homeomorphic

Can you give me some example that their fundamental group (which is non-trivial) is same but their topological spaces are not homeomorphic? $i.e$, \begin{align} \pi_1(X) = \pi_1 (Y), \qquad X \ncong ...
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1answer
15 views

Let $\omega_n: I \to S^1$ by $\omega_n(t)=\exp (2\pi int)$. Show that $\omega_p * \omega_q$ is homotopic, rel endpoints, to $\omega_{p+q}$.

Define $\omega_n: I \to S^1$ by $\omega_n(t)=\exp (2\pi int)$. Show that the concatenation $\omega_p * \omega_q$ is homotopic, rel endpoints, to $\omega_{p+q}$. (Consider first the special case of ...
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1answer
44 views

CW complex and equivalence of $[X,K(\mathbb{Z},n)]$ and $\langle X,K(\mathbb{Z},n)\rangle$

So Hatcher remarks that this is true when $X$ is connected and $n>0$, I was wondering if the result holds even if $X$ is not connected. If this isn't true what are some weaker assumptions we can ...
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1answer
15 views

Prove |[X,Z]| = |[Y,Z]| where X, Y are homotopic and Z is a top space

Since $X$ and $Y$ are homotopically equivalent there are two maps $f:X \to Y$ and $g:Y \to X$ which composites are homotopic to the appropiated identity maps. Now if I pick a representative of an ...
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1answer
18 views

Proving homotopy of 2 paths

$A$ and $B$ are $2$ points of affixes $i$ and $1$ respectively, $O$ is the origin. $γ_1=[AO]∪[OB]$ and $γ_2=[AB]$ are $2$ paths. I know how to prove that $2$ paths are homotopy but in this case I ...
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0answers
53 views

Homotopy of boundary paths

Let $G$ be a bounded, simply connected, open set in $\mathbb{R}^2$, and let $\gamma_1$ and $\gamma_1$ denote two paths such that $\gamma_0(0)=\gamma_1(0)$, $\gamma_0(1)=\gamma_1(1)$, and the following ...
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1answer
92 views

Can we always find homotopy of two paths which lies “between” the paths?

Let $\gamma_0,\gamma_1:[0,1]\to\mathbb{R}^2$ be paths such that $\gamma_0(0)=\gamma_1(0)$ and $\gamma_0(1)=\gamma_1(1)$. I wish to show that there is a homotopy ...
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2answers
41 views

$X$ is contractible if and only if $X \simeq \{ * \} $ - A three-part question

First some background: The topological spaces, $X, Y$, are homotopically equivalent if and only if there are continuous functions, $f \colon X \longrightarrow Y$ and $ g \colon Y \longrightarrow X $ ...
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1answer
10 views

Showing $\bar{p} * \alpha * p \simeq_{x_0} \bar{p} * \beta * p$, path-homotopy

For given path-connected topological spaces $X$, $x_0 , x_1 \in X$, and given loops $\alpha$, $\beta$ $: I \rightarrow X$ with base point at $x_1$ and a path $p: I \rightarrow X$ such that $p(0) = ...
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1answer
39 views

Homotopy of induced homomorphism

What i want to do is prove homotopy \begin{align} f \circ (\alpha * \beta) \simeq_{\{ f(x_0)\}} (f \circ \alpha) * ( f \circ \beta) \end{align} where $\circ$ is a composition $f \circ \alpha = ...
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0answers
30 views

Compute directly that the mapping cone of a homotopy equivalence is contractible

Let's consider the category $Ch_R$ of cochain complexes of modules over a commutative ring $R$. I'm trying to prove that if the chain map $\phi:M\rightarrow N$ is a homotopy equivalence then its ...
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1answer
81 views

Is the special orthogonal group really rationally homotopy commutative?

It is a classical result that the rational cohomology of $SO(n)$ is given by: $$H^*(SO(2m); \mathbb{Q}) = \begin{cases} S(\beta_1, \dots, \beta_{m-1}, \alpha_{2m-1}) & n = 2m \\ S(\beta_1, \dots, ...
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0answers
64 views

Can some Lie groups ($S^3$ in particular) be converted to simplicial groups?

I'm trying to understand the concept of Kan simplicial sets and simplicial groups in particular. My question is, in a nutshell: What is the relation between the group operation in a simplicial ...
3
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2answers
47 views

Winding number of composition of maps

If $f,g:S^{1}\rightarrow S^{1}$ maps, show that $N(f\circ g)=N(f)N(g)$, where $N(f)$ is the winding number of $f$. We defined the winding number of $f$ to be $N(f)=\frac{1}{2\pi}(\tilde{f}(1)- ...
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1answer
29 views

Model structure induced by a combinatorial model category.

In Hirschhorn's Model categories and their localizations he gives a sufficient condition to induce a cofibrantly generated model structure on a category $\mathcal{N}$, given an adjoint pair of ...
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0answers
37 views

Proving exactness in homotopy exact sequence

I am trying to work through Hatcher's proof (page 344 of his book) that the homotopy sequence of a triple $$... \rightarrow \pi_n(A,B,x_0) \stackrel{i_*}{\rightarrow} \pi_n(X,B,x_0) ...
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1answer
37 views

Square is homotopy Cartesian if horizontal maps are weak equivalences

This is probably trivial but I'm not the best with category theory. Let $M$ be a right proper model category (that is pullbacks of weak equivalences along fibrations are weak equivalences). The ...
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33 views

How to find fundamental groups and covering spaces of $\mathbb{RP}^2\vee \mathbb{RP}^2$?

The following is an exercise I was assigned in homotopy theory. Defined $X = \mathbb{RP}^2\vee \mathbb{RP}^2$. a) Find $\pi_1(X)$. b) Find the universal cover of $X$. c) Find all of its connected ...
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2answers
44 views

Homology of $\mathbb R^n \times \mathbb R^n \setminus \Delta_{\mathbb R^n \times \mathbb R^n}$

Here $\Delta_{\mathbb R^n \times \mathbb R^n}= \{(x,y) \in \mathbb R^n \times \mathbb R^n \mid x=y\}$. My idea was that for $n=1$ we have $$\mathbb R \times \mathbb R \setminus \Delta_{\mathbb R ...
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1answer
86 views

Isomorphism between homotopy groups of Lie group, Grassmann manifold

It is asserted without proof in a book edited by Novikov and Rokhlin that $$\pi_{k - 1}(\text{GL}_n^+(\mathbb{R})) \cong \pi_k(\tilde{G}_n).$$ I know how to show that these two spaces are bijective. ...
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0answers
17 views

Homotopy equivalence of a shrinked full torus

My task is to find $n,k$ such that $ S^{n} \vee S^{k} $ is homotopy equivalent to $[S^{1} \times D^{2}] /S^{1} \times S^{1} $ .By lookig at $S^{1} \times D^{2}$ as an "full" torus I've figured ...
4
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1answer
219 views

A map of a compact connected manifold with non-empty boundary to a sphere of the same dimension is null-homotopic

In Milnor & Kervaire's Groups of Homotopy Spheres paper, this claim: A map of a compact connected manifold with non-empty boundary to a sphere of the same dimension is null-homotopic is made ...
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1answer
51 views

Unique way to show $S^n$, $n \geq 2$ is simply connected.

This questions is asked in Armstrong's Topology book, and I am totally stuck.... I could really use a major hint: Think of $S^n \subset \mathbb{E}^{n+1}$. Given a loop $\alpha \in \pi_1 (S^n , ...
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1answer
15 views

any continuous function is null homotopic for convex set.

Let $X$ be a topological space. and suppose $B$ is a convex subset in $\mathbb{R}^n$. Prove that any continuous map $f: X \rightarrow B$ is null-homotopic. My strategy is following the defintion of ...
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1answer
47 views

Example of building a classifying space

I'm reading some things about algebraic topology, and they mention the classifying space of a group $G$ as $BG$, but they doesn't build one, so I want to ask if someone knows where can I find the way ...
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1answer
22 views

How to show the constant path is the identity element in fundamental group?

Let $X$ be a topological space and $q$ is a point in $X$. Denote the fundamental group of $X$ based at $q$ by $\pi _1(X,q)$. Then how should I verify that the constant path $c_q(s)\equiv q$ is the ...
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2answers
45 views

For which values of $k$ is there an $X$ with $\Omega^kX \cong X$?

Bott periodicity can be formulated as $\Omega^2 U \cong U$ where $\Omega$ denotes the based loop space functor and $U$ is the direct limit of unitary groups. The real version can be formulated as ...
3
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1answer
41 views

Classification of line bundles by group homomorphisms from the fundamental group to $\mathbb{Z}_2$

Let $X$ be a path-connected manifold. Then by the classification of vector bundles, the collection of all (real) line bundles on $X$ is $$ \text{Vect}^1(X)\cong ...
2
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1answer
34 views

Is it true that $\pi_n(X^{n-1})=0$ for $n>1$ if $X$ is $K(G,1)$ space?

In this post, Joe Johnson 126 mentioned the above fact, which I'm skeptical of. It is well-known that $\pi_n(X^{n+1})=\pi_n(X)$, but being a $K(G,1)$ space doesn't seem to imply the identity in the ...
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18 views

Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
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1answer
78 views

Cobordism and h-cobordism

Is there a way to simply explain cobordism and h-cobordism? I am not looking for a math based explanation, but rather, just the main ideas behind the concepts.
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1answer
65 views

Why is the group $[\Sigma\Sigma X, Y]_{\ast}$ commutative?

Can anyone give a reference (or explain here), why the group $[\Sigma\Sigma X,Y]_*$ is commutative? How is it related to the fact that $\Sigma X$ is a co-H-space?
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16 views

What is the topology of uniform convergence in this case of $P=P(x_{0},M)$ of all paths in $M$ starting at $x_{0}$?

The following definition I found it in a text on Lie groups: Let $M$ be a connected smooth manifold and $x_{0}\in M$. A path in $M$ starting at $x_{0}$ is a continuous curve $\gamma ...
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1answer
32 views

free commutative graded algebra

Let $V$ be a free graded module. $\wedge V$ be the free commutative graded algebra. $\wedge V$ = symmetric algebra $(V^{even})\otimes$ exterior algebra $(V^{odd})$ I don't understand this equation ...
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0answers
16 views

Formulation of zeroth order deformation equation (HAM)

I am reading the paper by Liao 'An explicit, totally analytic approximate solution for Blasius’ viscous flow problems', although this question equally applies to the other formulations of the homotopy ...
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1answer
112 views

How to show that a map without fix point from annular region to annular region is homotopic to antipodal map

$\Omega=\{x\in R^3: 1\le||x||\le2\}$ If $L:\Omega\rightarrow \Omega $ is continuous and without fix point , how to show $L$ is homotopic with antipodal map $x\rightarrow -x$?
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26 views

What is the suspension of the integers?

I'm interested to know the homotopy type of the reduced suspension of the integers. I suspect this should be a countable wedge of spheres, by extending the usual picture that indicates the ...
6
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2answers
75 views

Relation between two notions of $BG$

The following is something that's always niggled me a little bit. I usually think about stacks over schemes, so I'm a bit out of my element—I apologize if I say anything silly below. Let $G$ be a ...
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1answer
38 views

When does triviality of $f_*$ imply null-homotopy of $f$?

I've been considering problems of the following type: Given certain topological spaces $X$ and $Y$ and a continuous function $f:X\to Y$, prove that $f$ is null-homotopic. The cases studied so ...
2
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1answer
69 views

A category whose classifying space has nontrivial higher homotopy groups

The classifying space of a category $\scr{C}$ is obtained by taking its nerve $N\scr{C}$, which is the simplicial set defined by $$ N\mathscr{C}_n:= \mathrm{Fun}([n],\mathscr{C}) $$ and the ...
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0answers
13 views

Properties of a basic construction on co-H-spaces

Suppose $Y$ is a co-H-space. If $Y$ is standard, in the sense of Iwase, then $Y$ admits a homotopy decomposition $Y\simeq X\vee S$ where $X$ is simply-connected and $S$ is a wedge of circles. If $Y$ ...
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1answer
47 views

Definition of the infinite unitary group

I'm studying Suslin's proof of Bott Periodicity for the infinite unitary group $U$ which I currently understand to be $$\bigcup_{n\in\mathbb{N}}U(n)$$ where U(n) denotes the group of $n\times n$ ...
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1answer
36 views

How to show there is no homotopy between two curves?

On $\mathbb{C}\setminus\{0\}$, it seems that there can't be a homotopy between the curve given by $e^{i\theta},0\leq\theta<2\pi $ and the curve given by $e^{i\theta}+10,0\leq\theta<2\pi $. But ...
3
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1answer
36 views

Geometric intuition for homotopy invariance of fiber bundles?

There's a nice result in algebraic topology saying that given a fiber bundle, its pullbacks along homotopic maps are isomorphic as bundles. Thinking of a bundle as a comb with the "teeth" as its ...
2
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1answer
80 views

Fiber bundles that can be turned into a fibration that is a fiber bundle.

Let me recall a standard construction. Up to homotopy equivalence, any map $f: X \to Y$ is a fibering. Take the special case where $X=E$ the total space of a fiber bundle, and $Y$=B, the basespace ...
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1answer
59 views

When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$.

Let $X$ be a connected CW complex. Let $X_n$ fit into a commutative postnikov diagram for $X$ and let the fibrations $K(\pi_n(X),n) \hookrightarrow X_n \xrightarrow{\mathscr p} X_{n-1}$ be given. ...
0
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0answers
19 views

Minimal axioms for $X_n$ to be nth element of postnikov tower

Mosher and Tangora motivate using postnikov towers by saying they want a space $X(n)$ that represents in a certain sense, $X$, through dimension $n$. I am teaching a lecture on postnikov towers ...