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.

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how to install the Bvph_2.0 package in mathematica.

actually, I'm not that much familiar with and I am struggling to install the Bvph_2.0 package in mathematica software to solve a problem by HAM. can anyone please guide me.
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Under which condition is $\pi_2 (\phi)$ a free $\mathbb{Z}\pi_1 (X)$-module?

‎For a given map $\phi‎ :‎X\longrightarrow Y$‎, ‎the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$‎. ‎Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\...
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Path homotopy is preserved under composition

Suppose $F_0,F_1: X \rightarrow Y$ and $G_0,G_1: Y \rightarrow Z$ are continuous maps. Suppose that $F_0 \simeq F_1$ and $G_0 \simeq G_1$, show then that $G_0 \circ F_0 \simeq G_1 \circ F_1$. First ...
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Cohomology operations are group homomorphisms

Let our spaces have the homotopy type of a CW complex, and let $E^*,F^*$ be two cohomology theories. A (degree $n$, stable) cohomology operation is a map $\Phi: E^q(X) \to F^{q+n}(X)$ for each $q$, ...
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Projection of $S^3$ on $S^5$ in their fiber bundle

I'm working on some physics research involving the topology of the group $SU(3)$, which has long been known to be a non-trivial fiber bundle of the unit spheres $S^5 \times S^3$, with $S^3$ the fibers....
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Criterion for homotopy invertibility

let $\mathcal{A}$ be a pre-triangulated dg-category in the sense of Bondal and Kapranov, i.e. the homotopy category $H^0(\mathcal{A})$ is triangulated. Let $\phi\colon A\rightarrow B$ be a zero degree ...
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Detecting homotopy by precomposing with paths.

Let X and Y be topological spaces, and denote $\mathbb{S}^n$ the $n$-sphere. i) Suppose that $f,g:X \to Y$ are maps such that for every path $\alpha: [0,1] \to X,$ we have that $f \circ \alpha$ is ...
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Separability of the Real Interval in Classical Homotopy Theory

Question: what (if any) results in classical homotopy theory rely on the separability of the real interval? For a more expanded, fleshed out, and (in my opinion) interesting version of the context ...
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Computing pullback of $E_1$-ring spectra?

How do we compute pullbacks of $E_1$-ring spectra? An example I have in mind is the following from Land-Tamme (example in section 4). They claim that this is a pullback of $E_1$-ring spectra, but I ...
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Reference request: A specific book on homotopy and covering spaces

Twenty five or so years ago I read a library book that was meant to be a gentle introduction to covering spaces. The basic idea was that given a connected surface, and given a point in the space, the ...
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The "connecting homomorphism" for the LES of iterated classifying spaces for an SES of topological Abelian groups

Assume we are working with a "nice" category $\mathrm{Top}$ of topological spaces closed under the categorical constructions we'll use. Let $B : \mathrm{TopGrp} \rightarrow \mathrm{Top}_*$ ...
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Extension of map $f:X \to Y$ to cone $CX$ when $X$ is compact subset of $\mathbb{R^n}$

Let $X$ subset of $\mathbb{R^n}$ compact and $f:X\to Y$. Let $w=(0,0..0,1)$ in $\mathbb{R^n}\times\mathbb R$ and $C(X)$ be the union of all line segments joining points of $X$ to $w$. To show that $f$ ...
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Explicit description of classifying space as a "delooping"?

Consider the "classifying-space functor" $\mathscr{B} : \mathrm{TopGrp} \rightarrow \mathrm{Top}$, constructed in the standard way as a geometric realization of a nerve (in TopCat if ...
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Reference request: A co-H-space is a retract of a suspension

I would like a reference for the following result: Let $X$ be a co-H-space. Then $X$ is a retract of a suspension $\Sigma Z$.
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Filling an outer horn in a quasicategory when the first edge or the last edge is an equivalence

I convinced myself some time ago that the following holds: given an outer horn in a quasicategory, if the first or last edge (the map $0\to 1$ or $n-1\to n$) is an equivalence then the horn has a ...
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Torus filled with disk

Let I have a torus $T^2$, I am taking two meridian circle say $m_1$ and $m_2$ now I am attaching boundry of two different disk with the meridian circle $m_1$ and $m_2$, my question is is it ...
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How to relate the definition of mapping cylinder from homological algebra to topology?

I'm now learning triangulated categories and I have to work with the Homotopy category. There are distinguished triangles presented by cones and cylinders. However, I cannot understand the definition ...
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Induced maps on homotopy groups of $SU(2) \rightarrow G$

We have $\pi_3(G) = \mathbb{Z}$ for all compact connected simple Lie groups, and we know that given a map $\phi: SU(2) \rightarrow G$, the induced map $\phi_{*} : \mathbb{Z} \rightarrow \mathbb{Z}$ on ...
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Reference for topological invariants written in integrals

When studying physics, I came up with various integrals that only takes integer values due to topological reasons. Winding number $S^1 \to S^1$ is the most elementary example, which moreover provides ...
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Is a classifying space of $S^1$ the $S^1$ itself?

I almost surely believe that it is not, but can't find a mistake in my argument below: Fix a manifold $M$ and consider orientable plane bundles. These have $GL^+(2,\mathbb{R})$ structure group, which ...
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Is every fibration fiber homotopy equivalent to a fiber bundle?

A fibration $p : E \to B $ over a contractible base B is fiber homotopy equivalent to a product fibration $B \times F \to B$. (Corollary 4.63. Hatcher's Algebraic Topology) A locally trivial bundle(or ...
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Notations for Whitehead tower in Anderson duality

In appendix B of Hopkins and Singer's paper, Lemma B.15., the authors claimed that we can deduce the following isomorphisms $$ [X, \Sigma^n \tilde{I}]\rightarrow [X \langle n-1, \infty\rangle , \Sigma^...
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Triangulation of Mobius Strip to find the fundamental group

I am trying to show from image below that the fundamental group of the Mobius strip is $\pi_1(\text{Mobius strip})\cong\mathbb{Z}$ by finding the maximal tree of the Mobius strip. I know how to do ...
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Critic for a proof of a part on problem Hatcher's 1.1.5

Show that if every map $S^1 \to X$ is homotopic to a constant map, then every map $S^1 \to X$ extends to a map $D^2 \to X$. I've seen a few solutions for this questions on the site, but none of which ...
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Show that the change-of-basepoint homomorphism $\beta_h$ depends only on the homotopy class of $h$.

Show that the change-of-basepoint homomorphism $\beta_h$ depends only on the homotopy class of $h$. The change-of-basepoint homomorphism is defined as $\beta_h:\pi_1(X, x_1) \to \pi_1(X,x_0)$ ...
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The monoidal structure on the fundamental groupoids of spectrum

I am trying to understand Anderson duality and Picard categories from appendix B of Hopkins and Singer's paper, and I somehow get stuck on Example B.7 (Page 87). For a spectrum $E$, they consider each ...
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Homotopy group of 2-skeleton

Let $X=T^n$ be an $n$-torus and consider $X^{(2)}$ its $2$-skeleton. I know that $\pi_1(X^{(2)})= \pi_1(X)= \mathbb{Z}^n$, but which are the other homotopy groups of $X^{(2)}?$ I guess that $\pi_n(X^{(...
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Proof of homotopy lifting lemma.

I am a graduate student.In this semester we have a topology course with a basic introduction to fundamental groups and covering spaces.While studying covering spaces,I encountered the following ...
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Hatcher's problem 0.6

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the horizontal segment $ [0,1]\times\{0\}$ together with the vertical segments $\{r\}\times[0,1-r]$ for $r$ a rational number in $[0,1]$. Show ...
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What's the intuition for weak homotopy equivalence?

Defintition: Two topological spaces $X$ and $Y$ are said to be homotopy equivalent if there exist continuous maps $f: X \to Y$ and $g: Y \to X$, such that the composition $f \circ g$ is homotopic to $\...
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Show that if $f_0 \simeq f_1 \text{ rel } A$ and $g_0 \simeq g_1 \text{ rel }B$, then $g_0 \circ f_0 \simeq g_1 \circ f_1 \text{ rel } A$.

Let $A \subset X$ and $B \subset Y$ and suppose that $f_0,f_1 : X \to Y$ with $f_0\mid_A = f_1\mid_A$ and $f_i(A) \subset B$ for $i=0,1.$ Assume also that $g_0,g_1:Y \to Z$ are such that $g_0\mid_B =...
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The cartesian product of contractible spaces is contractible [duplicate]

Let $X_i$, $i\in I$ be contractible spaces. Is the Cartesian product $\prod_iX_i$ contractible, too?
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Connected sum minus a point is homotopy equivalent to connected sum of spaces each with a point taken out?

I.e. if your spaces are $X_1, \dots, X_m$, then is $$ \bigl(X_1 \mathbin{\#} \cdots \mathbin{\#} X_m \bigr) \backslash \{p\} $$ homotopy equivalent to $$ \bigl(X_1 \backslash {p_1}\bigr) \mathbin{\#} \...
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Let $X = \{0\} \cup \{1, 1/2, 1/3, ... , 1/n, ... \}$ and let $Y$ be a countable discrete space. Show that $X$ & $Y$ don't have the same homotopy type

Let $X = \{0\} \cup \{1, 1/2, 1/3, ... , 1/n, ... \}$ and let $Y$ be a countable discrete space. Show that $X$ and $Y$ do not have the same homotopy type. I think that the author wants me to show a ...
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Homotopy type of realization of singular simplicial space

It's well known that geometric realization of a simplicial set is a CW complex. More over, let $X$ be a space, then the canonical counit $u:|SX|\rightarrow X$ is a weak homotopy equivalence, where $S:...
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Proving Brouwer's fixed point theorem using fundamental groups

I am writing my bachelor thesis on the fundamental group $\pi_1(X)$ and homotopy theory. Now I was wondering if it is possible to prove Brouwer's fixed point Theorem in arbitrary dimensions using only ...
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Are CW complexes closed under homotopy colimits?

Consider the category of topological spaces homotopy equivalent (in the strong sense) to CW complexes. Is this category closed under arbitrary homotopy colimits? how about filtered homotopy colimits? ...
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Group/H-space structures for standard models of classifying spaces?

Let $G$ be a commutative topological group. May in his textbook, A concise course in algebraic topology, gives a model of the classifying space $BG$ so that it is a commutative topological group ...
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Inverses in the spherical interpetration of higher homotopy groups

I have just started studying the higher homotopy groups $\pi_n(X, x_0)$ in more detail than I have before and I am getting confused about the inverse operation which makes $\pi_n(X, x_0)$ a group. I'...
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When is a subset is homotopy type of a regular covering?

Let $X$ and $Y$ are finite CW complexes such that $X\subsetneq Y$ such that $X$ is not homotopy equivalent to $Y$. Can $X$ be homotopy type of a finite regular covering of $Y$? If Yes can we have an ...
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Why is this map anodyne?

As written in the title, I have some trouble in showing that a specific map is anodyne. The map in question is the inclusion of $J$ inn $\Delta^n \times \Delta^1 \times \Delta^1$, where $J$ is the ...
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Is $f_1: \pi_1(X \vee X) \rightarrow \pi_1(X)$ a surjection?

Suppose we have a space $X$ Does the following always hold? Does there exist a map $f: \pi_1(X \vee X) \rightarrow \pi_1(X)$ which is surjective? More generally does Do there exist maps $f_1: \pi_1(...
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Confusions about function spectrum in Anderson duality

In appendix B of this paper https://arxiv.org/abs/math/0211216, Hopkins and Singer defined the Anderson dual $\tilde{I}(E)$ of a spectrum $E$ as the function spectrum of maps from $E$ to $\tilde{I}$, ...
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Does the proof that the homotopy groups are abelian in Switzer's book only work for hausdorff spaces?

The proof that the homotopy groups of a pointed space, $\pi_n(X,x_0)$, are abelian for $n\geq 2$, given by Robert P. Switzer in his book "Algebraic Topology - Homology and Homotopy", relies ...
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intuition behind the construction of the map for showing associativity of $\pi_1(X,x_0)$.

I am studying algebraic topology.I have started with the chapter on fundamental group.Fundamental group at $x_0$ is defined to be the set of all equivalence classes of loops based at $x_0$ together ...
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Must a certain continuous map have 0 in its image, given that its restriction to the unit sphere is homotopic to the identity?

Suppose $f:\mathbb{B}^n \to \mathbb{R}^n$ is continuous (here $\mathbb{B}^n$ refers to the $n$-dimensional unit ball). Suppose also that its restriction $g := f|_{\mathbb{S}^{n-1}}$ does not have $0$ ...
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deformation to the normal cone in homotopy purity

I'm reading the proof of homotopy purity theorem. A key step in the proof is \textbf{Deformation to the normal cone}: Let $Z\xrightarrow{i}X$ be a closed immersion in $Sm/S$, where $S$ is noetherian ...
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what is definition of homotopy of homeomorphisms and isotopy of homeomorphisms?

I read "A Primer on Mapping Class Groups" By Benson Farb, Dan Margalit. I can't find definition of homotopy of homeomorphisms and isotopy of homeomorphisms in this book but in this book we ...
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Is There a Smooth Approximation to Classifying Spaces $BG\,?$

If you look at https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism, right above contents the claim is made that we can approximate the classifying space by smooth manifolds. I am aware of at ...
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Homotopy equivalence of $BGL_n(\mathbb{R})$ and $BO_n(\mathbb{R})$

I have tried to prove the above thing. My idea was the following: $\iota:O_n(\mathbb{R})\to GL_n(\mathbb{R})$ be the inclusion map which is a group homomorphism. It induces a fibre bundle $B\iota:BO_n(...
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