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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Algebraic cycles and approximation theorems in real algebraic geometry


Authors: J. Bochnak and W. Kucharz
Journal: Trans. Amer. Math. Soc. 337 (1993), 463-472
MSC: Primary 57R19; Secondary 14C25, 14P25
MathSciNet review: 1091703
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Abstract: Let $ M$ be a compact $ {C^\infty }$ manifold. A theorem of Nash-Tognoli asserts that $ M$ has an algebraic model, that is, $ M$ is diffeomorphic to a nonsingular real algebraic set $ X$. Let $ H_{{\text{alg}}}^k(X,\mathbb{Z}/2)$ denote the subgroup of $ {H^k}(X,\mathbb{Z}/2)$ of the cohomology classes determined by algebraic cycles of codimension $ k$ on $ X$. Assuming that $ M$ is connected, orientable and $ \dim\,M \geq 5$, we prove in this paper that a subgroup $ G$ of $ {H^2}(M,\mathbb{Z}/2)$ is isomorphic to $ H_{{\text{alg}}}^2(X,\mathbb{Z}/2)$ for some algebraic model $ X$ of $ M$ if and only if $ {w_2}(TM)$ is in $ G$ and each element of $ G$ is of the form $ {w_2}(\xi )$ for some real vector bundle $ \xi $ over $ M$, where $ {w_2}$ stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups $ G$ of $ {H^1}(M,\mathbb{Z}/2)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1091703-8
PII: S 0002-9947(1993)1091703-8
Article copyright: © Copyright 1993 American Mathematical Society