Skip to main content

All Questions

Filter by
Sorted by
Tagged with
16 votes
1 answer
601 views

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–––...
John Klein's user avatar
  • 18.8k
12 votes
1 answer
1k views

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-...
Anthony's user avatar
  • 283
9 votes
1 answer
671 views

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 $\...
R. van Dobben de Bruyn's user avatar
9 votes
1 answer
392 views

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 $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
J.K.T.'s user avatar
  • 517
8 votes
1 answer
470 views

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 $X\vee Y$. ...
J.K.T.'s user avatar
  • 517
8 votes
1 answer
756 views

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,...
João Lobo Fernandes's user avatar
8 votes
0 answers
342 views

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 $\alpha\colon S^p\to G$ ($p\geq1$)...
Charles Rezk's user avatar
  • 27.2k
7 votes
0 answers
194 views

"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 ...
Matt's user avatar
  • 208
2 votes
0 answers
109 views

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] ...
Matt's user avatar
  • 208