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Denote by $V_{n,k}$ the set of $k$-frames in $\mathbb{R}^n$. This is a Stiefel manifold. I am trying to understand the following statement of the cellular structure (from Mosher & Tangora's book page 42)

By a normal cell of $V_{n,k}$ we mean a cell of the form $e^{i_1-1} \times \cdots \times e^{i_r-1} \to \mathbb{R}P^{i_1-1} \times \cdots \times \mathbb{R} P^{i_r-1} \to V_{n,k}$ with $n \ge i_1 > \cdots > i_r > n-k$

(Note that if you compare to the actual book, what Mosher & Tangora call $P_{i_1}$ is actually $\mathbb{R}P^{i_1-1}$)

The cells of $V_{n,k}$ are exactly the normal cells and the $0$-cell corresponding to the identity matrix

The first test for this theorem is when $k=1$, whence we see that we must have $i_1=n$ and $r=1$, and thus the only normal cell is $e^{n-1} \to \mathbb{R} P^{n-1} \to V_{n,1}$

This raises my first question: $V_{n,1}$ is just $S^{n-1}$ so, as expected we get a $0$-cell and an $n-1$-cell. What, if anything, is the fact that the cell has the form $e^{n-1} \to \mathbb{R} P^{n-1} \to V_{n,1}$ actually mean?

My second test case was to consider $V_{3,2}$. Now I think, but I am not 100% on this (since the labelling of Stiefel manifolds is very inconsistent between books), that this is just $SO(3) \simeq \mathbb{R} P^3$, which we know has 4 cells, 1 in each dimension from 0 to 3. Now to satisfy $3 \ge i_1 > \cdots > i_r > 1$ it is clear that $r=2, i_1=3,i_2=2$ and we have the normal cell $e^{2} \times e^{1} \to \mathbb{R}P^{2} \times \times \mathbb{R} P^{1} \to V_{3,2}$.

Again, how can I interpret this; is the term 'normal cell' a slight misnomer, in that it is actually (in this case) 2-cells? (I guess there is something wrong here since I should get 4 cells?) Again what does the map between the cells and the product of projective spaces actually mean? Is this giving the characteristic map of the CW complex?

(Are there other nice references for CW-structure/cohomology of Stiefel manifolds? There is only a little bit in Hatcher's book as well.)

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up vote 1 down vote accepted

First question: Since there are two cells in this cell structure, one $0$-cell and one $n-1$-cell, there aren't many choices for attaching maps: the entire boundary of $e_{n-1}$ must be mapped to the $0$-cell.

Second question: The sequences need not be maximal. That is, you need to also consider the sequences $\{3\}$ and $\{2\}$. These give you your $2$-cell and $1$-cell, respectively.

A good reference for this sort of thing is I. M. James' book 'The Topology of Stiefel Manifolds'.

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Thank you, that is exactly what I needed. I should have realised that! – Juan S Aug 18 '11 at 5:10

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