## 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

Zbl 0284.55020
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### References:

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