Questions tagged [homotopy-theory]
Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called a homotopy group can be obtained from the equivalence classes. The simplest homotopy group is the fundamental group. Homotopy groups are important invariants in algebraic topology.
4,734 questions
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Homotopy theories from model theoretic perspective
Usually, I've seen homotopy theory being done abstractly via category theory and model structures on model categories. But is there a way to do it model-theoretic, similarly how algebraic theories are ...
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$E$-Cohomology of the Space of Units of a Ring Spectrum
Let $R$ be a ring spectrum and $GL_1(R)$ its space of units. Let $E$ be another ring spectrum. I am interested in computing $E^\ast(GL_1(R))$ and $E^\ast(B^k GL_1(R))$ when $R$ is not the sphere ...
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A strong deformation retraction onto a submanifold which does not restrict within arbitrarily small neighborhoods of points in the submanifold
A strong deformation retraction of a space $X$ onto a subspace $A$ is a continuous map $F : X \times [0, 1] \to X$ such that $F(x, 0) = x$, $F(x, 1) \in A$, and $F(a, t) = a$ holds for all $x \in X$, $...
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Local homology of topological manifold at its chart $U$
Suppose $M$ is a topological manifold and $\dim M = n$. If $U$ is a chart of $M$,i.e. $U$ is homeomorphic to $\mathbb R^n$, and suppose $U \subsetneq M$,does $H_n(M,M-U)\cong \mathbb{Z}$ ? For local ...
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Showing (with elementary methods) that a 'coin slot' is not simply connected. [closed]
I have been trying to come up with an example of a set that can be shown to be connected, yet not simply connected with elementary methods. Having struggled too much with $\mathbb{R}^2\setminus{\{\...
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Extending homeomorphism from the boundary
I have been stuck on this, I would appreciate some hints! Suppose $f: B \times I \to C \times I$ ($B,C$: homeomorphic closed orientable surfaces, and $I$ is the unit interval) is such that: • $f(B \...
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Fundamental group of a complement of a certain graph (attempt 2)
NOTE: I initially tried to ask this question here but forgot to include an important detail from the exercise (that the space in question is different from the immersed Klein bottle in that an open ...
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Fundamental group of a complement of a certain graph
Hatcher's exercise 1.2.12 consists of a few parts. I was able to solve all of them except the last part, which asks the reader to show that the immersed Klein bottle in $\mathbb{R}^3$, shown in the ...
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Duality functors on the Spanier–Whitehead category
Let $\mathcal{S}\mathcal{W}$ be the Spanier–Whitehead category (the full subcategory of the stable homotopy category spanned by finite spectra). Let $D : \mathcal{S}\mathcal{W}^\mathrm{op} \to \...
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Discussion of Spanier–Whitehead duality in Adams' Stable Homotopy and Generalized Homology
In Adams' book (Section III.5), he describes the following construction, due to Spanier. Consider two finite simplicial complexes $K, L$ which are embedded disjointly in $S^n$ such that $L$ is a ...
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Can Homotopy Classes of functions from a Surface be classified by Homotopy Groups?
Suppose $M$ is a connected compact topological manifold. If you consider a closed connected 1-dimensional manifold $X$, then it's homeomorphic to a circle $X\cong \mathbb{S}^1$. Let's denote $[X,M]$ ...
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Morphism between the fibres of a fibration
Let $p:E\to B$ be a fibration. For $b\in B$, denote by $F_b$ the set $p^{-1}(b)$. Let $b, b'\in B$, and let $\gamma: I \to B$ be a path starting from $b$ and ending in $b'$. I want to define a map $\...
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4votes
1answer
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Why is the quotient category on objects not well-defined?
I am taking classes on Category theory, and my professor recently talked about weak equivalences, congruences on a category, quotient categories, category of fractions... Every one of those ...
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$X$ contractible and $Y$ arc connected then any two maps from $X$ to $Y$ are homotopic
I need to prove that if $X$ is a contractible space and $Y$ is arc-connected then any two applications from $X$ to $Y$ are homotopic. I tried to approach it like this, but I'm missing something. $X$ ...
3votes
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Nerve Theorem: explicit description of the homotopy equivalence.
This is the definition of the nerve of a open cover of a topological space: Let $X$ be a topological space, and let $\mathcal{U}$ be a cover of $X$ by open subsets. The nerve of $\mathcal{U}$, ...
