Inclusion of Grassmannians Induces Isomorphism on Cohomology

Question

Let $\text{Gr}_2^+(\mathbb{R}^n)$ denote the the Grassmannian of oriented $2$-planes in $\mathbb{R}^n$, i.e. the orientation-double-cover of the usual Grassmannian $\text{Gr}_2(\mathbb{R}^n)$. Let $i : \text{Gr}_2^+(\mathbb{R}^n) \hookrightarrow \text{Gr}_2^+(\mathbb{R}^\infty) \simeq \mathbb{CP}^\infty$ denote the canonical inclusion. I am trying to prove the following:

The inclusion $i$ induces isomorphisms $\mathbb{Z}[e] \cong H^*(\text{Gr}_2^+(\mathbb{R}^\infty)) \to H^*(\text{Gr}_2^+(\mathbb{R}^n))$ for all degrees $* < n-2$.

Using a homology-pair-LES argument together with Hurewicz (more specifically Hatcher, Theorem 4.32) it suffices to show that

1. $\text{Gr}_2^+(\mathbb{R}^n)$ is simply connected.
2. The pair $(\text{Gr}_2^+(\mathbb{R}^\infty), \text{Gr}_2^+(\mathbb{R}^n))$ is $(n-3)$-connected.

The first item is fine. For the second item I tried using Corollary 4.12 in Hatcher, i.e. I need to show that the pair is a CW-pair (an argument for this can be found in Hatcher's book on vector bundles) and that all cells in $\text{Gr}_2^+(\mathbb{R}^\infty) \setminus \text{Gr}_2^+(\mathbb{R}^n)$ have dimension $> n-3$. But this last step false in general I believe, since $\text{Gr}_2^+(\mathbb{R}^n)$ is not the $k$-skeleton of $\text{Gr}_2^+(\mathbb{R}^\infty)$ for any $k \in \mathbb{N}$, so in general this difference could still contain cells of dimension smaller than $n-3$. At least in the literature I have not found any statement about $\text{Gr}_2^+(\mathbb{R}^n)$ being some skeleton of $\text{Gr}_2^+(\mathbb{R}^\infty)$, even though it is possible to show some related properties; for instance that $\text{Gr}_2^+(\mathbb{R}^n)$ is a finite sub-complex of dimension exactly $2n-4$.

Question: Can I modify my arguments to give a coherent proof, or do I need a different proof altogether?

Remark 1: For reference, this problem can be found on p. 8 of Galatius' lecture notes on the Madsen-Weiss Theorem.

Remark 2: An alternative approach uses Hatcher Lemma 4D.2, namely if there exists an $(n-3)$-connected pair of spaces $(B,A)$ and a fibre bundle $p: \text{Gr}_2^+(\mathbb{R}^\infty) \to B$ with $p^{-1}(A) = \text{Gr}_2^+(\mathbb{R}^n)$, then the pair $(\text{Gr}_2^+(\mathbb{R}^\infty), \text{Gr}_2^+(\mathbb{R}^n))$ is also $(n-3)$-connected. However, so far I have not been able to come up with a pair satisfying these properties.

Remark 3: A third approach I can think of is to identify the quotient $\text{Gr}_2^+(\mathbb{R}^\infty) / \text{Gr}_2^+(\mathbb{R}^n)$ with some known space $X$ for which we have $H^k(X, *) \cong 0$ for all $k < n-2$, since we know that CW-pairs are good pairs. By a LES argument the original claim would follow. However I fail to come up with a suitable space $X$.

Any help or hints are appreciated!

Answer 1

There is nothing special about $2$-planes here, so let me be more general and consider $ \newcommand{\Gr}{\operatorname{Gr}} \newcommand{\R}{\mathbb{R}} $ $\Gr_k^+(\R^n)$, the space of oriented $k$-planes in $\R^n$. The corresponding statement is the following:

Proposition A: the embedding $i:\Gr_k^+(\R^n) \hookrightarrow \Gr_k^+(\R^{\infty})$ is $(n-k)$-connected.

As argued in the OP, this implies that $i$ induces an isomorphism in homology and cohomology in degrees $<n-k$ (for a reference, see for example Theorem 11.2 on p.481 of Bredon [1], or Corollary 6.69 on p.164 of Davis-Kirk [2]).

A standard tool to prove connectivity results such as Proposition A is to realize the spaces of interest as parts of a fiber bundle or fibration, and then study the associated long exact sequence of homotopy groups.

In the case of Grassmannians, it's useful to consider the following bundles, which appear in Example 4.53 of Hatcher [3]. Let $V_{k}(\R^n)$ be the Stiefel manifold of orthonormal $k$-frames in $\R^n$. Then we have a map $V_k(\R^n) \to \Gr_k(\R^n)$ sending a frame to the $k$-plane that it spans. This is a principal $O(k)$-bundle, where the $O(k)$ action is defined by "precomposing" a frame with an element of $O(k)$ (where we see a frame as an isometric embedding $\R^k \to \R^n$). In fact, this bundle is by definition the frame bundle of the tautological bundle of rank $k$ over $\Gr_k(\R^n)$.

Similarly, in the oriented case, we have a map $V_k(\R^n) \to \Gr_k^+(\R^n)$, which is defined as before but now the $k$-plane is equipped with the orientation determined by the frame. This is a principal $SO(k)$-bundle, and in fact it is the bundle of positively-oriented frames of the tautological bundle over $\Gr_k^+(\R^n)$.

This construction behaves naturally with respect to isometric embeddings $\R^n \hookrightarrow \R^m$, in the sense that such an embedding induces a diagram: $$ \tag{1}\label{cd:stabilization} \require{AMScd} \begin{CD} V_k(\R^n) @>>> V_k(\R^m)\\ @VVV @VVV \\ \Gr_k^+(\R^n) @>>> \Gr_k^+(\R^m) \end{CD}$$ which is a morphism of principal $SO(k)$-bundles.

The diagram \eqref{cd:stabilization} induces maps between the long exact sequences of homotopy groups of each bundle. By studying these maps, we will show:

Proposition B: The embedding $\Gr_k^+(\R^n) \hookrightarrow \Gr_k^+(\R^m)$ is $(n-k)$-connected.

Proof sketch: First, prove that the Stiefel manifold $V_k(\R^n)$ is $(n-k-1)$-connected. This computation can be found in many places: see for example p.202 of Whitehead [4] or Example 4.53 and 4.54 of Hatcher [3]. Again, the idea is to see $V_k(\R^n)$ as part of a fiber bundle. The proof in Hatcher uses the bundle $V_{k}(\R^n) \to S^{n-1}$ with fiber $V_{k-1}(\R^{n-1})$ that sends a $k$-frame to its last element.

Next, consider the map induced by Diagram \eqref{cd:stabilization} on the long exact sequences of homotopy groups of the bundles. Then, use the fact that the terms involving $V_k(\R^n)$ and $V_k(\R^m)$ all vanish in degrees $< n-k$ to conclude. $\square$

We can now deduce Proposition A:

Proof of Proposition A: We need a technical fact, which is that for a compact space $X$, any map $f: X \to \Gr_k^+(\R^{\infty})$ has its image in some finite-dimensional Grassmannian $\Gr_k^+(\R^N)$. This is proved for example in this MSE post.

In particular, any map $f: S^q \to \Gr_k^+(\R^{\infty})$ and any homotopy $H: S^q \times I \to \Gr_k^+(\R^{\infty})$ factors through a finite-dimensional Grassmannian. Combining this with the fact that the embedding $\Gr_k^+(\R^n) \hookrightarrow \Gr_k^+(\R^N)$ is $(n-k)$-connected, we can deduce that $i$ is also $(n-k)$-connected. $\square$

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References

[1] G. E. Bredon. Topology and Geometry. Springer, 1993.

[2] J. F. Davis and P. Kirk. Lecture notes in algebraic topology. AMS, 2001.

[3] A. Hatcher. Algebraic Topology. Cambridge University Press, 2002.

[4] G. W. Whitehead. Elements of Homotopy Theory. Springer, 1978.