Questions tagged [higher-homotopy-groups]

136 questions
17
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1 answer

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…
14
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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…
10
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1 answer

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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10
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1 answer

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: I wonder whether there are some useful information that I can relate $\pi_m(SO(N))$ and $\pi_m(O(N))$? Here is the difficulty somehow posted by MO to obtain $\pi_m(O(N))$,…
8
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1 answer

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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8
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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…
7
votes
0 answers

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 $(X',A',B')$. $f$…
7
votes
2 answers

$\pi_1(A,x_0)$ 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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6
votes
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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…
6
votes
1 answer

Is the splitting $\pi_{k}(X,A)\simeq\pi_{k}(X)\times \pi_{k-1}(A)$ a $\pi_1(A)$-modules isomorphism?

Let $(X,A)$ be a pair of topological spaces with $A\subset X$. Fix a basepoint $x_0$ of $X$ which lies in $A$. Assume that the inclusion $(A,x_0)\to (X,x_0)$ is homotopic to a constant map relatively to the basepoint, that is there is a homotopy…
6
votes
1 answer

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…
5
votes
1 answer

Does there exist a smooth compact manifold whose homotopy groups are not finitely generated?

Does there exist a smooth compact manifold whose homotopy groups are not finitely generated? I found a counter-example for topological manifolds, but I did not understand whether it is possible to introduce a smooth structure on it
5
votes
1 answer

How torsion arise in homotopy groups of spheres?

The example I have in mind of a torsion element in fundamental group is what happens on the projective plane: there is a cell $D^2$, but its boundary doubly covers a copy of $S^1$. The latter has then order two in the homotopy group. Let me call…
5
votes
2 answers

homotopy groups of product space

I'm trying to prove that the sequence below is exact $1 \to \pi_{n}(X) \to \pi_{n}(X \times Y) \to \pi_{n}(Y)$ Such that $i_{*}:\pi_{n}(X) \to \pi_{n}(X \times Y)$ and $\pi_{y*}:\pi_{n}(X \times Y) \to \pi_{n}(Y)$ How do I show that…
5
votes
1 answer

How do synthetic homotopy groups relate to the usual homotopy groups?

In Homotopy Type Theory (HoTT in what follows) one may compute homotopy groups of objects that bear names that are the same as some usual spaces: for instance one may consider $S^1$ which is defined by $*:S^1, b : *=_{S^1}*$ and with the usual…
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