Topological manifolds and real algebraic geometry

Alberto Tognoli

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 3, page 545-555
  • ISSN: 0392-4033

Abstract

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We study the problem of approximating, up to homotopy, compact topological manifolds by real algebraic varieties. As a consequence, we realize any integral non-degenerate quadratic form as the intersection form of a real algebraic variety. This is related to a well-known result, due to Freedman [F], on the topology of closed simply-connected topological 4 -manifolds.

How to cite

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Tognoli, Alberto. "Topological manifolds and real algebraic geometry." Bollettino dell'Unione Matematica Italiana 6-B.3 (2003): 545-555. <http://eudml.org/doc/194830>.

@article{Tognoli2003,
abstract = {We study the problem of approximating, up to homotopy, compact topological manifolds by real algebraic varieties. As a consequence, we realize any integral non-degenerate quadratic form as the intersection form of a real algebraic variety. This is related to a well-known result, due to Freedman [F], on the topology of closed simply-connected topological $4$-manifolds.},
author = {Tognoli, Alberto},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {545-555},
publisher = {Unione Matematica Italiana},
title = {Topological manifolds and real algebraic geometry},
url = {http://eudml.org/doc/194830},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Tognoli, Alberto
TI - Topological manifolds and real algebraic geometry
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/10//
PB - Unione Matematica Italiana
VL - 6-B
IS - 3
SP - 545
EP - 555
AB - We study the problem of approximating, up to homotopy, compact topological manifolds by real algebraic varieties. As a consequence, we realize any integral non-degenerate quadratic form as the intersection form of a real algebraic variety. This is related to a well-known result, due to Freedman [F], on the topology of closed simply-connected topological $4$-manifolds.
LA - eng
UR - http://eudml.org/doc/194830
ER -

References

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  1. AKBULUT, S.- KING, H., The topology of real algebraic sets with isolated singularities, Ann. of Math., 113 (1981), 425-446. Zbl0494.57004MR621011
  2. BENEDETTI, R.- RISLER, J. J., Real algebraic and semialgebraic sets, Hermann, Paris (1990). Zbl0694.14006MR1070358
  3. BERETTA, L., Extension of maps and ordering, Ann. Univ. Ferrara, 44 (1998), 9-25. Zbl0952.54010MR1744138
  4. CAVICCHIOLI, A., Sulla classificazione topologica delle varietà, Boll. Un. Mat. Ital. (7), 9-B (1995), 633-682. Zbl0857.57019
  5. CAVICCHIOLI, A.- HEGENBARTH, F.- SPAGGIARI, F., A splitting theorem for homotopy equivalent smooth 4 -manifolds, Rendiconti di Matematica Ser. VII, 17 (1997), 523-539. Zbl0890.57032MR1608716
  6. CURTIS, C. L.- FREEDMAN, M. H.- HSIANG, W. C.- STONG, R., A decomposition theorem for h -cobordant smooth simply-connected compact 4 -manifolds, Invent. Math., 123 (1996), 343-348. Zbl0843.57020MR1374205
  7. DONALDSON, S., An application of gauge theory to four-dimensional topology, J. Differential Geom., 18 (1983), 269-315. Zbl0507.57010MR710056
  8. DONALDSON, S., Connections, cohomology and the intersection forms of 4 -manifolds, J. Differential Geom., 24 (1986), 275-341. Zbl0635.57007MR868974
  9. DONALDSON, S., The orientation of Yang-Mills moduli spaces and four-dimensional topology, J. Differential Geom., 26 (1987), 397-428. Zbl0683.57005MR910015
  10. DONALDSON, S., Irrationality and the h -cobordism conjecture, J. Differential Geom., 26 (1987), 141-168. Zbl0631.57010MR892034
  11. FREEDMAN, M. H., The topology of four-dimensional manifolds, J. Differential Geom., 17 (1982), 357-453. Zbl0528.57011MR679066

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