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Questions tagged [higher-homotopy-groups]

For questions related to higher homotopy groups. A higher homotopy group, $\pi_n(X,x_0)$, is the set of based homotopy classes of based maps $\gamma:(S^n,s_0)\rightarrow(X,x_0).$

The fundamental group, $\pi_1(X,x_0)$, is a powerful tool of algebraic topology used to study topological spaces by studying how loops behave within them. Higher homotopy groups, which will be our main object of study, are the natural generalization of the fundamental group.

While $\pi_1(X,x_0)$ considers how loops can live in a space up to based homotopy, higher homotopy groups more generally consider how closed surfaces (maps from the $n$-sphere) can be mapped into spaces.

A higher homotopy group, $\pi_n(X,x_0)$, is the set of based homotopy classes of based maps $\gamma:(S^n,s_0)\rightarrow(X,x_0).$

As in the case of the fundamental group, higher homotopy groups are also groups.

202 questions
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Homotopy groups U(N) and SU(N): $\pi_m(U(N))=\pi_m(SU(N))$

Am I correct that homotopy groups of $U(N)$ and $SU(N)$ are the same, $$\pi_m(U(N))=\pi_m(SU(N)), \text{ for } m \geq 2$$ except that $$\pi_1(U(N))=\mathbb{Z}, \;\;\pi_1(SU(N))=0,$$ Hence the Table in page 3 of this note has error along the column…
wonderich
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Whitehead product and a homotopy group of a wedge sum

Note : this question has been crossposted on the mathematics Overflow. Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-complex. My goal is to prove the following isomorphism : $$\pi_{n+k+1}(X\vee…
Anthony
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Are the higher homotopy groups of a compact manifold finitely generated as $\mathbb{Z}[\pi_1]$-modules?

Let $M$ be a compact manifold. The homology and cohomology groups of $M$ are necessarily finitely generated, as is the fundamental group. Serre proved that a simply connected finite CW complex has finitely generated homotopy groups, so if $M$ is…
Michael Albanese
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Homotopy groups O(N) and SO(N): $\pi_m(O(N))$ v.s. $\pi_m(SO(N))$

I have known the data of $\pi_m(SO(N))$ from this Table: $$\overset{\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\textbf{Homotopy groups of orthogonal groups}}{\begin{array}{lccccccccc} \hline & \pi_1 & \pi_2 & \pi_3 & \pi_4 & \pi_5 &…
wonderich
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Can a simply connected manifold satisfy $M\simeq M\times M?$

Let $M$ be a simply connected, (finite dimensional) smooth manifold. Is it possible that $M$ is homotopy equivalent to $M\times M,$ without $M$ being contractible? This would imply $\pi_n(M)\times\pi_n(M)\cong \pi_n(M)$ fo all $n\in\mathbb{N}.$ I…
JLA
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Confusion about free homotopy, based homotopy and homotopy groups

Unfortunately, this becomes a very general post: I have some questions concerning the homotopy invariance of homotopy groups. I start from what should be clear: If $f,g:(X,x_0)\to (Y,y_0)$ are based homotopic, then $\pi_k(f)=\pi_k(g)$, which is…
FKranhold
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What does fundamental (homotopy) groups measure?

As I read an algebraic topology book, I felt I knew exactly what the fundamental group is geometrically! I thought it counts the number of independent cycles. (my definition of dependence cycles (that may be incorrect): $\alpha,\beta$ are two…
C.F.G
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A possible error in May's Concise Algebraic Topology (prespectra)

In chapter 22.1 of May's A Concise Course in Algebraic Topology, he claims that the prespectrum $\{T_n\}$ of spaces where each $T_n$ is $(n-1)$-connected yields a reduced homology theory by setting $\tilde{E}_q(X) = \operatorname{colim}_n…
e_s_
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Why is the Whitehead product $[\iota_2, \eta] : \pi_4\mathbb{S}^2$ trivial?

Let $\iota_2 : \pi_2\mathbb{S}^2$ be a generator and let $\eta : \pi_3\mathbb{S}^2$ be the Hopf map. The Whitehead product $[\iota_2, [\iota_2, \iota_2]] : \pi_4\mathbb{S}^2$ must be trivial, because $$[\iota_2, [\iota_2, \iota_2]] = [\iota_2,…
Tom
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Show that the space $Y = S^3 \vee S^6$ has precisely two distinct homotopy classes of comultiplications.

Here is the question: A comultiplication for a pointed space $X$ is a map $\phi : X \rightarrow X \vee X$ so that the composite $$X \xrightarrow{\phi} X \vee X \xrightarrow{i_{X}} X \times X$$ is homotopic to the diagonal map. Show that the space $X…
Intuition
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Do Homotopy Groups commute with generalized filtered colimits?

I know that if $X$ is a topological space such that $X= \underset{i}{\bigcup} X_i$ where $X_0 \subset X_1 \subset ... \subset X_n \subset ...$, where $X_i$ are all hausdorff, then the functor $\pi_n(\_)$ commutes with the colimit: $$\varinjlim…
Balletti
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Five lemma for homotopy exact sequences of triple

Suppose we have topological spaces $B\subset A\subset X$ and $B'\subset A'\subset X'$, and a continuous map $f:X\to X'$ with $f(A)\subset A'$, $f(B)\subset B'$. Consider the homotopy long exact sequence of the triples $(X,A,B)$ and $(X',A',B')$. $f$…
user302934
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$\pi_1(A,x_0)$ acts on the long exact sequence of homotopy groups for $(X,A,x_0)$

In the last paragraph in page 345 of Hatcher's Algebraic Topology(link:http://pi.math.cornell.edu/~hatcher/AT/ATch4.pdf), Hatcher says that $\pi_1(A,x_0)$ acts on the long exact sequence of homotopy groups for $(X,A,x_0)$, the action commuting with…
blancket
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Nullhomotopicity of $\mathbb{S}/p \to^p \mathbb{S}/p$ for $p=2$ and $p \neq 2$?

For a given spectra $X$ we have $X/p$ defined as the cofiber $X \to^p X$ where the map is basically defined via defining it on the sphere spectrum $\mathbb{S}$! To define $\cdot p$ on the sphere spectrum we just need to define it on $\mathbb{S^1}$…
user135743
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Homotopy cardinality of fiber bundles

Consider the Homotopy cardinality (or $\infty$-groupoid cardinality) $\chi(X):=\sum_{[x]\in\pi_0(X)}\prod_{i\geq0}|\pi_i(X,x)|^{(-1)^{i+1}}$ associated to a space $X$. Suppose we have a fibre bundle $F\to E\stackrel{p}{\to}B$ such that for each…
deeppurp
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