A general image computing spectral sequence. (English) Zbl 1141.55011
Chéniot, Denis (ed.) et al., Singularity theory. Proceedings of the 2005 Marseille singularity school and conference, CIRM, Marseille, France, January 24–February 25, 2005. Dedicated to Jean-Paul Brasselet on his 60th birthday. Singapore: World Scientific (ISBN 978-981-270-410-8/hbk). 651-675 (2007).
This paper is a contribution to the problem of computing the homology of the image of a mapping, \(f: X\to Y\). The author generalizes methods of V. Goryunov and D. Mond [Compos. Math. 89, No. 1, 45–80 (1993; Zbl 0839.32017)] to derive a spectral sequence converging to \(H_*(f(X); G)\) for \(G\) any Abelian group of coefficients. The main tool is a simplicial resolution of the space \(Y\) using multiple point spaces,
\[ D^k(f) = \text{ closure}\,\{ (x_1, \dots, x_k) \in X^k\mid x_i \neq x_j \text{ if } i\neq j, f(x_1) = \cdots = f(x_k)\}. \]
The collection \(\{D^k(f)\mid k \geq 1\}\) enjoy face-maps, \(\varepsilon_{k,i}: D^k(f) \to D^{k-1}(f)\) given by forgetting the \(i\)th entry. There is also a mapping \(\varepsilon_k : D^k(F) \to Y\) given by \(\varepsilon_k(x_1, \dots, x_k) = f(x_j)\), for any \(j\). Let \(M_k(f) = \varepsilon_k(D^k(f))\). The face complex \((D^\bullet(f), \varepsilon_{\bullet, i})\), as a simplicial space, has a realization which gives a space \(W\) together with a mapping \(g: W \to Y\) which is a homotopy equivalence when \(Y\) and \(W\) are finite complexes. The space \(W\) has a filtration by skeleta, and for a reasonable mapping \(f: X\to Y\) (for example, there is a finite bound on the cardinality of the preimage of any point in \(Y\)), the filtration is complete.
The action of the symmetric groups on the multiple point spaces allow one to identify points in the image with orbits. For another group action and \(f: X\to Y\) an equivariant mapping with respect to the \(G'\)-action, under reasonable conditions, there is a spectral sequence with
\[ E^1_{p,q} = H_q^{\text{alt}\langle S_{p+1}\rangle\times G'}(D^{p+1}(f); G) \]
and converging to \(H^{G'}_*(f(X);G)\), the \(G'\)-equivariant homology of the image. The author generalizes further to include relative versions of the spectral sequence. Details about the equivariant nature of the construction are given explicitly. This version of the construction includes the previous work of the author and Goryunov.
For the entire collection see [Zbl 1111.14001].
\[ D^k(f) = \text{ closure}\,\{ (x_1, \dots, x_k) \in X^k\mid x_i \neq x_j \text{ if } i\neq j, f(x_1) = \cdots = f(x_k)\}. \]
The collection \(\{D^k(f)\mid k \geq 1\}\) enjoy face-maps, \(\varepsilon_{k,i}: D^k(f) \to D^{k-1}(f)\) given by forgetting the \(i\)th entry. There is also a mapping \(\varepsilon_k : D^k(F) \to Y\) given by \(\varepsilon_k(x_1, \dots, x_k) = f(x_j)\), for any \(j\). Let \(M_k(f) = \varepsilon_k(D^k(f))\). The face complex \((D^\bullet(f), \varepsilon_{\bullet, i})\), as a simplicial space, has a realization which gives a space \(W\) together with a mapping \(g: W \to Y\) which is a homotopy equivalence when \(Y\) and \(W\) are finite complexes. The space \(W\) has a filtration by skeleta, and for a reasonable mapping \(f: X\to Y\) (for example, there is a finite bound on the cardinality of the preimage of any point in \(Y\)), the filtration is complete.
The action of the symmetric groups on the multiple point spaces allow one to identify points in the image with orbits. For another group action and \(f: X\to Y\) an equivariant mapping with respect to the \(G'\)-action, under reasonable conditions, there is a spectral sequence with
\[ E^1_{p,q} = H_q^{\text{alt}\langle S_{p+1}\rangle\times G'}(D^{p+1}(f); G) \]
and converging to \(H^{G'}_*(f(X);G)\), the \(G'\)-equivariant homology of the image. The author generalizes further to include relative versions of the spectral sequence. Details about the equivariant nature of the construction are given explicitly. This version of the construction includes the previous work of the author and Goryunov.
For the entire collection see [Zbl 1111.14001].
Reviewer: John McCleary (Poughkeepsie)
MSC:
| 55T05 | General theory of spectral sequences in algebraic topology |
