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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49 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)- \tilde{...
1
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1answer
31 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 ...
1
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0answers
40 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) \stackrel{j_*}{\...
1
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1answer
39 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 ...
4
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0answers
37 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
91 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. ...
0
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0answers
18 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
votes
1answer
222 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 ...
0
votes
1answer
52 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 , ...
2
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1answer
17 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 ...
1
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1answer
58 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
23 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
46 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
votes
1answer
54 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 [X,BO(1)]=[X,\mathbb{R}P^\infty]=[X,B\...
2
votes
1answer
38 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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0answers
19 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 ...
1
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1answer
85 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.
5
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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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0answers
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 :[0,1]\...
0
votes
1answer
33 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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votes
0answers
18 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 ...
2
votes
1answer
113 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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0answers
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
votes
2answers
88 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 ...
1
vote
1answer
41 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
votes
1answer
72 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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votes
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 $Y$ ...
4
votes
1answer
48 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 $n\times n$ ...
1
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1answer
38 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
votes
1answer
37 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
votes
1answer
82 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
63 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
votes
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 ...
0
votes
1answer
55 views

$S^n/S^k$ is homotopy equivalent to $S^n\vee S^{k+1}$.

How can I prove that $ S^{n}/S^{k} $ is homotopy equivalent to $S^{n} \vee S^{k+1} $? Here $S^{n} \vee S^{k+1} $ is a space $ \left[ S^{n} \cup S^{k+1} \right]/\approx $ where $ \approx$ is an ...
4
votes
1answer
69 views

Calculate $\pi_2(S^2 \vee S^1)$

I am trying to calculate $\pi_2(S^2 \vee S^1)$ and having trouble fitting the pieces together. I know that the universal cover of $S^2 \vee S^1$ is just $\mathbb{R}$ with spheres attached at integral ...
2
votes
1answer
70 views

Hatcher's proof of Van Kampen Theorem

I am studying Van Kampen Theorem using Hatcher's textbook. I am dealing with the general statement, I mean: (pg 43) He defines previously the free product of groups (pg 41) as: I can follow the ...
0
votes
1answer
25 views

In R2\(axis X)the two nullhomotopic functions are not homotopic

I want to prove that in space R2\X the two functions f&g are not homotopic if we define g(x)=c1& f(x)=c2 c1 and c2 are two points one of the above axis X and the othe one is under it
2
votes
1answer
59 views

$C_\infty$ analog of the correspondence between $A_\infty$-alg. structures on $A$ and dg coalg. strucures on $(\bar T(sA),\Delta)$

There is a 1-1-correspondence between $A_\infty$-algebra structures on a graded vector space $A$ and dg. coalgebra structures on the bar construction $(\bar T(sA),\Delta)$. My question: Is there any ...
2
votes
1answer
57 views

about properties of homeomorphism

we know that the following shapes are homeomorphic but in my book write : Geometrically speaking, a homeomorphism is a bijection that can bend, twist, stretch, and wrinkle the space M to make it ...
2
votes
1answer
65 views

Universal covering space VS fibration from contractible total space

For a path-connected space $X$, a covering space is a fiber bundle with a discrete set. It is known that if $X$ in addition locally path-connected and semilocally simply-connected, then $X$ has a ...
2
votes
1answer
50 views

Multiplicative structure on algebraic K-theory

Let $R$ be a commutative ring. Using Quillen's $+$-construction, it is relatively easy to see that the algebraic K-theory of $R$, $K_*(R)$, admits a graded commutative product $$K_i(R)\otimes K_j(R) \...
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votes
0answers
29 views

Spectrum of spectrum in the stable homotopy category

Let $\mathcal{E}=(E_0,E_1,\cdots)$ be an $S^1$-spectrum. Define $\Sigma \mathcal{E}$ to be the spectrum with $(\Sigma \mathcal{E})_n=E_{n+1}$. Then, consider the spectrum $\tilde{\mathcal{E}} =(\...
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0answers
41 views

Expositions of Postkinov Towers

I am giving a lecture on Postkinov towers and I want to teach the students a lesson :). These students have seen local coefficients, spectral sequences and cohomology operations at the level of ...
0
votes
1answer
33 views

$1_{S^{n-1}} \simeq$ to a constant map

I have to show that $ \exists\ \ f : D^n \rightarrow S^{n-1} $ with $f\circ i =1_{S^{n-1}} \iff 1_{S^{n-1}} $ is homotopic to a constant map. I don't know how to prove this. So, please help me in ...
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votes
0answers
37 views

What does formality of a chain complex mean topologically?

I've been told that every topological abelian group is a product of Eilenberg-Mac Lane spaces, but I don't have a reference for this fact. This confuses me because via the Dold-Kan correspondence, ...
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1answer
31 views

Define a map $\Omega \Sigma \Omega Y \to \Omega Y$

Let $Z$ be $\Omega Y$ which is the space of loops based at $Y_0$. Then I know how to define a map explicitly from $Z \to \Omega \Sigma X$. It is defined by noting we have the identity map $ \Sigma Z,...
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vote
2answers
46 views

Show that $f$ is homotopic to $g$. [closed]

Let $X$ be any topological space. If $f,g:X\to S^n$ (n-sphere) are continuous, such that $f(x)$ and $g(x)$ are never antipodal, show that $f$ is homotopic to $g$. I have no idea about this, please ...
7
votes
1answer
80 views

Fibration: if map $i^*: H^*(X, G) \to H^*(F, G)$ is surjective, then action of $\pi_1(B)$ on $H^*(F, G)$ trivial?

For a fibration $F \overset{i}{\to} X \overset{p}{\to} B$ with $B$ path-connected, if the map $i^*: H^*(X, G) \to H^*(F, G)$ is surjective, then does it necessarily follow that the action of $\pi_1(B)$...
0
votes
1answer
51 views

Homotopy category of a simplicial category

In many places (for example here) I've seen the following definition: For a simplicial category $\mathcal{C}$, it's homotopy category is defined to be the category $Ho(\mathcal{C})$ with the same ...