an:06121486
Zbl 1276.55012
Kozlowski, A.; Yamaguchi, K.
Simplicial resolutions and spaces of algebraic maps between real projective spaces
EN
Topology Appl. 160, No. 1, 87-98 (2013).
00312453
2013
j
55P10 55P35
simplicial resolution; homotopy equivalence; Vassiliev spectral sequence
Let \(1\leq m< n\) be positive integers and define \(D_*(d;m,n)=(n-m)(\lfloor \frac{d+1}{2}\rfloor+1)-1\) and \(D(d;m,n)=(n-m)(d+1)-1\), for each integer \(d\geq 0\). In this paper the authors improve some results from [\textit{M. Adamaszek} and the present authors, Q. J. Math. 62, No. 4, 771--790 (2011; Zbl 1245.14060)] replacing \(D_*\) by \(D\). More precisely, they show that \(\tilde{A}_d(m,n)\) has the same homology (resp. homotopy) as the mapping space \(\mathrm{Map}(\mathbb{R}P^m,\mathbb{R}P^n)\) up to dimension \(D(d;m,n)\) if \(m+1=n\) (resp. \(m+2\leq n\)), where \(\tilde{A}_d(m,n)\) is the projectivization \(A_d(m,n)(\mathbb{R})/\mathbb{R}^*\) of the space of all \((n+1)\)-tuples \((f_0,\ldots,f_n)\in \mathbb{R}[z_0,\ldots,z_m]^{n+1}\) of homogeneous polynomials of degree \(d\) without non-trivial common real roots. Also, Theorem~1.5 provides some tools to generalize some results concerning the space of algebraic maps from real projective spaces to complex projective spaces. This is done in the authors' paper [Spaces of equivariant algebraic maps from real projective spaces into complex projective spaces; \url{arXiv:1109.0353} [math.AT]].
The authors use spectral sequences induced from truncated resolutions to obtain the main results.
Thiago de Melo (Rio Claro)
Zbl 1245.14060