3
$\begingroup$

I was going through this paper by Tanaka but I am stuck at Proposition 4.1 given below enter image description here. I just cannot make sense of the first two lines of the proof. What does it mean when he says S-reducible and reducible? I tried searching online also but could not find anything, nor does the author refer to anything.

Can someone help me with this?

Thanks and regards in advance.

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ I've never heard that term, but given the context, could it just mean "the $n$-manifold $M$ is S-reducible if the identity of $S^n$ factors through $M$ up to homotopy"? $\endgroup$
    – Najib Idrissi
    Commented Nov 4, 2022 at 12:31
  • $\begingroup$ @NajibIdrissi If that's the case we need to see if the same is true for 3-dim projective space? What maps can we consider on and from sphere through the projective space $\endgroup$
    – Devendra Singh Rana
    Commented Nov 4, 2022 at 17:45

1 Answer 1

Reset to default
5
$\begingroup$

An $n$-dimensional CW complex with a single $n$-cell is reducible if the projection $X \to X/X^{(n-1)} = S^n$ onto the top cell admits a section up to homotopy. It is stably reducible, or S-reducible, if such a section exists for the associated suspension spectra. An early use of this term is in

James, I. M.
Spaces associated with Stiefel manifolds.
Proc. London Math. Soc. (3) 9 (1959), 115–140.

The dual notions (coreducible and S-coreducible) appear in

Atiyah, M. F.
Thom complexes.
Proc. London Math. Soc. (3) 11 (1961), 291–310.

These ideas play a role in Adams' solution of the vector fields on spheres problem.

Adams, J. F.
Vector fields on spheres.
Ann. of Math. (2) 75 (1962), 603–632.
Improve this answer
$\endgroup$
2
  • $\begingroup$ Can you explain " a section up to homotopy". @JohnRognes $\endgroup$
    – Devendra Singh Rana
    Commented Nov 7, 2022 at 6:59
  • $\begingroup$ @DevendraSinghRana Please consult the references I gave, or a textbook, such as Husemoller's "Fibre Bundles" or Hatcher's "Algebraic topology". $\endgroup$
    – John Rognes
    Commented Nov 7, 2022 at 8:56

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .