Algebra, Algebraic Topology and their Interactions Proceedings of a
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Algebra, Algebraic Topology and their Interactions Proceedings of a Conference held in Stockholm, Aug. 3-13, 1983, and later developments
Edited by J.-E. Roos
Springer-Verlag Berlin Heidelberg New York Tokyo
Editor
Jan-Erik Roos Department of Mathematics, University of Stockholm Box 6701, 11385 Stockholm, Sweden
Mathematics Subject Classification (1980): 13-06, 13D03, 13E05, 13H99, 13J10, 14-06, 14F35, 16A24, 17B70, 18G15, 18G20, 20F05, 20F10, 55-06, 55P35, 55015, 55S30, 57-xx ISBN 3-540-16453-7 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16453-7 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
A MATHEMATICAL INTRODUCTION These notes contain the outcome and later developments ar1s1ng from a Nordic Summer th , 1983 on "ALGEBRA,
School and Research Symposium held in Stockholm, August 3-13 ALGEBRAIC TOPOLOGY AND THEIR INTERACTIONS".
Let me first give a brief indication of the ma1n ideas behind this symposium. During the last decade several striking analogies between algebraic topology (at least rational homotopy theory) and algebra (at least local algebra) had been observed. Let me just give two examples. (More examples and details can be found in the paper Through the looking glass: A dictionary between rational homotopy
and local
algebra by L. AVRAMOV and S. HALPERIN in these proceedings.) First some preliminaries. Let X be a finite, simply-connected CW-complex, of loops on X and
the rational homology algebra of
a Hopf algebra.) At the same time, let
be a local commutative noetherian ring
R with maximal ideal m and residue field k = vector space
the space
(This algebra is even
and let Ext;(k,k) be the graded
equipped with the algebra structure coming from the Yoneda
@
n>O
composition
0
----->
Ext;(k,k) is also a Hopf algebra and
This Yoneda Ext-algebra is even the enveloping algebra of a certain
graded Lie algebra TI*(R) over k. On the other hand, it is also known that the enveloping algebra of the rational homotopy Lie algebra Samelson product on this Lie algebra corresponds under the isomorphism to the Whitehead product on the
is
(Note that the n
(X)
(X).) We are now ready for the examples:
Example 1.- Let F -----> E -----> B be a Serre fibration and ••.- - > TI
n+ 1(B)
a
- - > TIn(F) --> TIn(E) - - > TIn(B) --->
(1)
the corresponding homotopy exact sequence. In [7] Halperin proved (under some minor extra conditions) that, if H*(F,@) is finite dimensional, then (1) breaks up into exact seguen
