Questions tagged [whitehead-product]
A product on the homotopy groups of a space: $\pi_k(X)\times\pi_l(X) \to \pi_{k+l-1}(X)$.
10 questions
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Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?
Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
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Non-triviality of Whitehead products in wedges of CW-complexes
Suppose $X$ and $Y$ are finite, simply connected, based CW-complexes and $m,n\geq 2$. If $a\in \pi_m(X)$ and $b\in \pi_n(Y)$, then one can regard these as elements of the homotopy groups of $X\vee Y$. ...
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9
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Homotopy groups of finite CW complex finitely generated as Lie algebra
This is probably a well-known question, but I haven't found the answer on MO or MSE.
It is well-known that the homotopy groups of a finite CW complex $X$ need not be finitely presented, even as $\...
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The image of the J-homomorphism of the tangent bundle of the sphere
Consider the J-homomorphism $\pi_{n-1}(SO(n))\to \pi_{2n-1}(S^n)$ and $\tau_{S^n}:S^{n-1}\to SO(n)$ to be the adjoint of the classifying map of the tangent bundle of the standard sphere. In this paper,...
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Sum of higher Whitehead products
For a CW-complex $X$ and homotopy classes $[f_i] \in \pi_{l_i}(X)$ for $1 \leq i \leq s$, where $s \geq 2$, the higher Whitehead product is defined to be the set
$$[f_1,...,f_s] := \{[\rho \circ \phi] ...
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Whitehead product and a homotopy group of a wedge sum
Note : this is a crosspost from the Mathematics StackExchange, as suggested by this meta post.
Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-...
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"Relative Whitehead products"
The notion of a relative Whitehead product exists in the literature and has been asked about before (e.g. here). I am trying to find out about a different product on relative homotopy classes which ...
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CW Product via Whitehead map
Product CW-complexes are defined via characteristic maps rather than from attaching maps, so via maps from $\mathbb D^n$ rather than from $\mathbb S^{n-1}$, because we have the propriety that $\mathbb ...
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Whitehead products and Framed Manifolds
The attaching map for the top cell of the torus $S^n \times S^n$ is a map
$$
[x,y]: S^{2n-1} \to S^n \vee S^n
$$
where the notation is such that
$x,y : S^n \to S^n \vee S^n$ are the two inclusions–––...
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Reference request: Whitehead product and the Borel construction
This is a question about signs.
Fix
a based space $(X,x_0)$,
a topological group $G$
acting on $X$ from the left, so that the basepoint $x_0$ is fixed,
a based map $\alpha\colon S^p\to G$ ($p\geq1$)...
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