# Tagged Questions

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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### Covering spaces of wedge sum of circles

Let's suppose I'd like to characterise all connected covering spaces of wedge sum of circles.When it comes to a torus, a sphere or $S^{1} \vee S^{2}$ it is easy to point out exactly how do all ...
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### Homotopy classes of manifolds of dimension $2n$ which are $(n-1)$ connected

In his essay "Classification of (n − 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres" Milnor states (in passing) "The manifold can be built up (up to homotopy type) by taking ...
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### Characteristic function is an identification function

Every characteristic function $\Phi_{\beta}^b: E^n_{\beta} \to e^{-n}_{\beta}$ is an identification function. My book says the following: This follows from the fact the the CW complex $X$ has $X^n$ ...
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### Annulus Homotopic to punctured plane

I know a circle is homtopic to a punctured plane, and by the same reasoning, the aannulus must also be, as it a "step" in the homotopy (IE the annulus is a "stretched" circle). The only trouble is it ...
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### What is the map from $H_j( \Sigma MSO(k)) \to H_{j-k}(BSO(k))$ on Tom Dieck page 537

I am reading Tom Dieck's page 537 and I am not sure what the vertical map that I put in the title is in the diagram in the bottom of the page. This map is labeled Thom Isomorphism. Here $MSO(k)$ is ...
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### IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$?

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$? $\pi(X)$ denotes the fundamental group of $X$.
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### Prove homotopy equivalence of two spaces

How can I prove that $[S^{1} \times D^{2}]/S^{1} \times S^{1}$ is homotopy equivalent to $S^{2} \vee S^{3}$? So far I have proved that $S^{2} \vee S^{3} \cong S^{3}/S^{1}$ Additionaly applying ...
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### 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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### 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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### 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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### 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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### $X$ 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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### 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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It is a classical result that the rational cohomology of $SO(n)$ is given by: $$H^*(SO(2m); \mathbb{Q}) = \begin{cases} S(\beta_1, \dots, \beta_{m-1}, \alpha_{2m-1}) & n = 2m \\ S(\beta_1, \dots, ... 0answers 64 views ### Can some Lie groups (S^3 in particular) be converted to simplicial groups? I'm trying to understand the concept of Kan simplicial sets and simplicial groups in particular. My question is, in a nutshell: What is the relation between the group operation in a simplicial ... 2answers 49 views ### Winding number of composition of maps If f,g:S^{1}\rightarrow S^{1} maps, show that N(f\circ g)=N(f)N(g), where N(f) is the winding number of f. We defined the winding number of f to be N(f)=\frac{1}{2\pi}(\tilde{f}(1)- \tilde{... 1answer 31 views ### Model structure induced by a combinatorial model category. In Hirschhorn's Model categories and their localizations he gives a sufficient condition to induce a cofibrantly generated model structure on a category \mathcal{N}, given an adjoint pair of ... 0answers 39 views ### Proving exactness in homotopy exact sequence I am trying to work through Hatcher's proof (page 344 of his book) that the homotopy sequence of a triple$$... \rightarrow \pi_n(A,B,x_0) \stackrel{i_*}{\rightarrow} \pi_n(X,B,x_0) \stackrel{j_*}{\...
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 ...