Simplicial resolutions and spaces of algebraic maps between real projective spaces

We show that the space A ˜ d ( m , n ) consisting of all real projective classes of ( n + 1 ) -tuples of real coefficients homogeneous polynomials of degree d in ( m + 1 ) variables, without common real roots except zero, has the same homology as the space Map ( R P m , R P n ) of continuous maps from the m-dimensional real projective space R P m into the n real dimensional projective space R P n up to dimension ( n − m ) ( d + 1 ) − 1 . This considerably improves the main result of Adamaszek et al. (2011) [1].