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On the homotopy type of complete intersections. (English) Zbl 1072.14063
Let $${\mathbf d}=(d_1,\dots ,d_k)$$ be an unordered $$k$$-tuple, $$X_n({\mathbf d}) \subset {\mathbb C}\mathbb P^{n+k}$$ a complete intersection i.e. the transversal intersection of $$k$$ hypersurfaces defined by vanishing of homogeneous polynomials of degrees $$d_1,\dots ,d_k.$$ Let $$d=d_1d_2\cdots d_k$$ be the total degree of $${\mathbf d}.$$ Natural questions that arise are the classification problems for $$X_n({\mathbf d}).$$ The authors study the homotopy classification problem i.e. when two complete intersections have the same homotopy type. For $$d$$ which does not have small divisors and $$n$$ odd, A. Libgober and J. Wood [Topology 21, 469–482 (1982; Zbl 0504.57015)] and F. Fang for $$n$$ even [Comment. Math. Helv. 72, No. 3, 466–480 (1997; Zbl 0896.14028)] have proved that homotopy type of $$X_n({\mathbf d})$$ is determined by $$n, d,$$ Euler characteristic and signature of $$X_n({\mathbf d}).$$ The authors solve this classification problem for $$n$$ large relative to $$d.$$ For a natural number $$k$$ let $${\nu}_p(k)$$ be the highest exponent of $$p$$ that divides $$k.$$ If $${\mathbf d}=(d_1,\dots,d_k)$$ define its $$p$$-localization to be the sequence obtained from $$(p^{{\nu}_p(d_1)},\dots ,p^{{\nu}_p(d_k)})$$ by deleting entries with $${\nu}_p(d_i)=0.$$ One calls $${\mathbf d}$$ to be equivalent to $${\mathbf {d'}}$$ if for all $$p$$ their $$p$$-localizations are the same. The authors prove the following theorems.
Theorem 1.1. If $${\mathbf d}$$ and $${\mathbf {d'}}$$ are equivalent, then for any $$n>2$$ the complete intersections $$X_n({\mathbf d})$$ and $$X_n({\mathbf {d'}})$$ are homotopy equivalent provided their Euler characteristics and signatures agree.
Theorem 1.2. If $${\mathbf d}$$ and $${\mathbf {d'}}$$ are nonequivalent multidegrees, there is a positive integer $$N({\mathbf d},{\mathbf {d'}})$$ such that if $$n\geq N({\mathbf d},{\mathbf {d'}})$$ then the complete intersections $$X_n({\mathbf d})$$ and $$X_n({\mathbf {d'}})$$ are not homotopy equivalent.
The authors derive from these theorems two interesting corollaries one of which says when (under some additional hypotheses on $$n, {\mathbf d}$$ and $${\mathbf {d'}}$$) the complete intersections $$X_n({\mathbf d})$$ and $$X_n({\mathbf {d'}})$$ are homeomorphic.

##### MSC:
 14M10 Complete intersections 55P15 Classification of homotopy type
##### Keywords:
homotopy classification problem; homeomorphism
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