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.


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


Zbl 0284.55020
Full Text: DOI Euclid


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