Weakly Semialgebraic Spaces (Lecture Notes in Mathematics)

Knebusch, Manfred

ISBN 10: 3540508155 ISBN 13: 9783540508151

Editorial: Springer, 1989

Idioma: Ingl�s

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Detalles bibliogr�ficos

T�tulo

Weakly Semialgebraic Spaces (Lecture Notes in Mathematics)

Editor

Springer

Fecha de publicaci�n

1989

Idioma

Ingl�s

ISBN 10

3540508155

ISBN 13

9783540508151

Encuadernaci�n

Paperback

Estado

Very Good

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Sinopsis

The book is the second part of an intended three-volume treatise on semialgebraic topology over an arbitrary real closed field R. In the first volume (LNM 1173) the category LSA(R) or regular paracompact locally semialgebraic spaces over R was studied. The category WSA(R) of weakly semialgebraic spaces over R - the focus of this new volume - contains LSA(R) as a full subcategory. The book provides ample evidence that WSA(R) is "the" right cadre to understand homotopy and homology of semialgebraic sets, while LSA(R) seems to be more natural and beautiful from a geometric angle. The semialgebraic sets appear in LSA(R) and WSA(R) as the full subcategory SA(R) of affine semialgebraic spaces. The theory is new although it borrows from algebraic topology. A highlight is the proof that every generalized topological (co)homology theory has a counterpart in WSA(R) with in some sense "the same", or even better, properties as the topological theory. Thus we may speak of ordinary (=singular) homology groups, orthogonal, unitary or symplectic K-groups, and various sorts of cobordism groups of a semialgebraic set over R. If R is not archimedean then it seems difficult to develop a satisfactory theory of these groups within the category of semialgebraic sets over R: with weakly semialgebraic spaces this becomes easy. It remains for us to interpret the elements of these groups in geometric terms: this is done here for ordinary (co)homology.

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