Localized at an odd prime there is a space $B_k$ which sits in a fibration $S^{2k+2p-3}\rightarrow B_k \rightarrow S^{2k-1}$ and has homology $H_{\ast}(B_k;\mathbb{Z}/p\mathbb{Z})\cong \Lambda(x_{2k-1},x_{2k+2p-3})$, the exterior algebra on two generators. I know that James wrote about these and showed some of their properties, but my question is about their construction. They can be constructed at a pull back, but how is this done? It isn't entirely clear to me and if any one has any suggestions they would be greatly appreciated!
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2$\begingroup$ I suppose you have i) a restriction on $k$ as well as ii) some condition on the homology (like $P^1(x_{2(k-1)})=x_{2k+2p-3})$ for example. Otherwise the product of the two spheres also has the properties you mention. $\endgroup$– user43326Commented Apr 14, 2014 at 13:25
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2 Answers
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Let's consider general $S^n$ fibrations over a space $X$; to make my life easy, I will assume that they come with a section (this is not a huge restriction). If $hAut(S^n)$ denotes the grouplike monoid of self-homotopy equivalences of $S^n$ that preserve the basepoint, then the set of isomorphism classes of sectioned $S^n$-fibrations over $X$ are in bijection with the set of homotopy classes of maps
$$[X, B hAut(S^n)]$$
from $X$ into the classifying space of $hAut(S^n)$. This is a fairly general phenomenon; $S^n$ can be replaced by any reasonable based space $Y$.
However, for $S^n$ the space of homotopy automorphisms is a familiar one. Since any map $S^n \to S^n$ of degree $\pm 1$ is a homotopy automorphism,
$$hAut(S^n) = \{f:S^n \to S^n, \mbox{ based, degree $\pm 1$} \} = \Omega^n_{\pm 1} S^n$$
is the $\pm 1$ set of components of the iterated loop space $\Omega^n S^n$.
To get the fibration that you're after, that would be classified by a map in
$$[S^{2k-1}, BhAut(S^{2k+2p-3})] = [S^{2k-2}, hAut(S^{2k+2p-3})] = [S^{2k-2}, \Omega^{2k+2p-3}_{\pm 1} S^{2k+2p-3}] = \pi_{4k+2p-5}(S^{2k+2p-3})$$
Unfortunately, this homotopy group is not in the stable range, so I don't know how to compute it; perhaps someone who knows more metastable homotopy than me will jump in here. Consequently, I can't say exactly how to construct this fibration.
However, there is a related construction that kind of fits the numerology of your question. For each odd prime $p$, the image of the $J$-homomorphism in the stable homotopy groups of spheres includes an element $\alpha \in \pi_{2p-3}^S(S^0)$ of order $p$. For sufficiently large $n$, this can be realized as a map $S^{n+2p-3} \to S^{n}$. I can form the adjoint map $S^{2p-3} \to \Omega^n S^n$. Notice that since this is based map and $S^{2p-3}$ is connected, its image is in the degree 0 component of $\Omega^n S^n$. Shift it by 1 so that it's image is the degree 1 component; this is the space of based homotopy self-equivalences of $S^n$ of degree 1, $hAut_1(S^n)$. The result is an element
$$\alpha \in [S^{2p-3}, hAut(S^n)] = [S^{2p-2}, B hAut(S^n)]$$
for large $n$, defining a nontrivial $S^n$-bundle over $S^{2p-2}$.
answered Apr 14, 2014 at 16:10
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2$\begingroup$ I should add: the fact that $\alpha$ is in the image of $J$ means that this fibration is in fact the fibrewise 1-point compactification of a vector bundle. $\endgroup$ Commented Apr 14, 2014 at 16:34
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1$\begingroup$ More explicitly still: if $H$ is the tautological bundle over $S^2 = \mathbb{C} P^1$, then $H-1$ trivializes over $\infty$. Thus the $(p-1)$-fold external tensor power $(H-1)^{\otimes p-1}$ over $(S^2)^{\times p-1}$ descends to a virtual bundle over $(S^2)^{\wedge p-1} = S^{2p-2}$. Adding $n$ copies of the trivial bundle for large $n$ will eventually result in an actual vector bundle; the sphere bundle described above is the fibrewise 1-point compactification of this vector bundle. $\endgroup$ Commented Apr 14, 2014 at 18:45
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Perhaps one should not change the question but I think that the order of the dimensions of the spheres is reversed. Assuming this, then a good place to start is the paper by Mimura and Toda Topology 9 (1970) 627-680, "Mod p decomposition of compact Lie groups 1. They describe these bundles as pullbacks from the unit tangent sphere bundle over S^2k. These spaces and other similar pull=backs have been studied extensively in the finite H-space literature. and can be trace by looking at the citations in Math Reviews of the Mimura-Toda paper.
