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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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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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 ...
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 $... 2answers 44 views Homology of$\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 $$\mathbb R \times \mathbb R \setminus \Delta_{\mathbb R \... 1answer 95 views Isomorphism between homotopy groups of Lie group, Grassmann manifold It is asserted without proof in a book edited by Novikov and Rokhlin that$$\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. ... 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 ... 1answer 267 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 ... 1answer 54 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 , ... 1answer 19 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 ... 1answer 59 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 ... 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 ... 2answers 47 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 \... 1answer 55 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\... 1answer 39 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 ... 1answer 30 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 ... 1answer 86 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. 1answer 66 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? 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]\...
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 ...