## Spanier-Whitehead duality in étale homotopy

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- by Roy Joshua PDF
- Trans. Amer. Math. Soc.
**296**(1986), 151-166 Request permission

## Abstract:

We construct a $(\bmod {\text {-}}l)$ Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a $(\bmod {\text {-}}l)$ Spanier-Whitehead dual.## References

- J. F. Adams,
*On the groups $J(X)$. II*, Topology**3**(1965), 137–171. MR**198468**, DOI 10.1016/0040-9383(65)90040-6 - M. Artin and B. Mazur,
*Etale homotopy*, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR**0245577**, DOI 10.1007/BFb0080957 - J. C. Becker and D. H. Gottlieb,
*Transfer maps for fibrations and duality*, Compositio Math.**33**(1976), no. 2, 107–133. MR**436137** - Paul Baum, William Fulton, and Robert MacPherson,
*Riemann-Roch and topological $K$ theory for singular varieties*, Acta Math.**143**(1979), no. 3-4, 155–192. MR**549773**, DOI 10.1007/BF02392091 - A. K. Bousfield and D. M. Kan,
*Homotopy limits, completions and localizations*, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573**, DOI 10.1007/978-3-540-38117-4 - Kenneth S. Brown,
*Abstract homotopy theory and generalized sheaf cohomology*, Trans. Amer. Math. Soc.**186**(1973), 419–458. MR**341469**, DOI 10.1090/S0002-9947-1973-0341469-9 - David A. Cox,
*Algebraic tubular neighborhoods. I, II*, Math. Scand.**42**(1978), no. 2, 211–228, 229–242. MR**512271**, DOI 10.7146/math.scand.a-11749
E. Friedlander, - Eric M. Friedlander,
*Étale homotopy of simplicial schemes*, Annals of Mathematics Studies, No. 104, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR**676809** - Eric M. Friedlander,
*Étale $K$-theory. I. Connections with etale cohomology and algebraic vector bundles*, Invent. Math.**60**(1980), no. 2, 105–134. MR**586424**, DOI 10.1007/BF01405150 - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157**, DOI 10.1007/978-1-4757-3849-0
R. Joshua, - Roy Joshua,
*Becker-Gottlieb transfer in étale homotopy*, Amer. J. Math.**109**(1987), no. 3, 453–497. MR**892595**, DOI 10.2307/2374564
—, $(\bmod {\text {-}}l)$ - Daniel M. Kan,
*Semisimplicial spectra*, Illinois J. Math.**7**(1963), 463–478. MR**153017** - Daniel M. Kan and George W. Whitehead,
*The reduced join of two spectra*, Topology**3**(1965), no. suppl, suppl. 2, 239–261. MR**178463**, DOI 10.1016/0040-9383(65)90077-7 - John W. Milnor and James D. Stasheff,
*Characteristic classes*, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR**0440554**, DOI 10.1515/9781400881826 - James S. Milne,
*Étale cohomology*, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR**559531** - E. H. Spanier,
*Function spaces and duality*, Ann. of Math. (2)**70**(1959), 338–378. MR**107862**, DOI 10.2307/1970107 - Robert M. Switzer,
*Algebraic topology—homotopy and homology*, Die Grundlehren der mathematischen Wissenschaften, Band 212, Springer-Verlag, New York-Heidelberg, 1975. MR**0385836**, DOI 10.1007/978-3-642-61923-6
R. Thomason,

*Fibrations in etale homotopy*, Inst. Hautes Études Sci. Publ. Math.

**42**(1971), 281-322.

*Geometric fibers for maps of simplicial schemes*, 1984, preprint. —,

*Thom spaces of algebraic vector bundles*(to appear).

*Spanier-Whitehead duality and Becker-Gottlieb transfer in etale homotopy*, Ph.D. thesis, Northwestern Univ., 1983.

*Riemann-Roch for algebraic vs. topological*$K$-

*theory*, preprint, 1981.

## Additional Information

- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**296**(1986), 151-166 - MSC: Primary 14F35; Secondary 55P25
- DOI: https://doi.org/10.1090/S0002-9947-1986-0837804-4
- MathSciNet review: 837804