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The homotopy of spaces of maps between real projective spaces. (English) Zbl 1112.55008

Write \(\text{Map}_1({\mathbb R} P^m, {\mathbb R} P^n)\) (respectively, \(\text{Map}^*_1({\mathbb R} P^m, {\mathbb R} P^n)\)) for the path component of the inclusion \({\mathbb R} P^m \hookrightarrow {\mathbb R} P^n\) (\(m \leq n\)) in the space of all (respectively, all basepoint preserving) continuous functions. Following the approach of S. Sasao in the complex case [J. Lond. Math. Soc. (2) 8, 193–197 (1974; Zbl 0284.55020)], the author computes the homotopy groups of these function spaces through a large range of degrees. The method is as follows: The right action of the orthogonal group \(O(n)\) on the space \(\text{Map}^*_1(\mathbb R P^m, \mathbb R P^n)\) gives rise to a map \(\alpha_{m,n} \colon V_{n,m} \to\text{Map}_1^*(\mathbb R P^m, \mathbb R P^n)\) where \(V_{n,m} = O(n)/O(n-m)\) is the real Stiefel manifold. Similarly, an action of \(O(n+1)\) on \(\text{Map}_1(\mathbb R P^m, \mathbb R P^n)\) gives rise to a map \(\beta_{m,n} \colon PV_{n+1, m+1} \to\text{Map}_1(\mathbb R P^m, \mathbb R P^n)\) where \(PV_{n+1, m+1} = O(n+1)/ \Delta_{m+1} \times O(n-m)\) and \(\Delta(m+1) \subset O(m+1)\) is the center. When \(m < n,\) the author proves that the maps \(\alpha_{m,n}\) and \(\beta_{m,n}\) induce isomorphisms on homotopy groups up to dimension \(2(n-m)-1\). This result leads to a complete calculation of the rational homotopy groups of these function spaces. The author also computes the fundamental groups when \(m=n\). The proofs are based on comparing various fibrations involving real Stiefel manifolds to fibrations arising from evaluation and restriction maps in the context of function spaces.

MSC:

55P15 Classification of homotopy type
55P62 Rational homotopy theory
55P10 Homotopy equivalences in algebraic topology

Citations:

Zbl 0284.55020
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References:

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