Obstructions to homotopy equivalences |
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Authors: | Stephen Halperin James Stasheff |
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Institution: | Department of Mathematics, Scarborough College University of Toronto, West Hill, Ontario M1C 1A4, Canada;Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27514 USA |
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Abstract: | An obstruction theory is developed to decide when an isomorphism of rational cohomology can be realized by a rational homotopy equivalence (either between rationally nilpotent spaces, or between commutative graded differential algebras). This is used to show that a cohomology isomorphism can be so realized whenever it can be realized over some field extension (a result obtained independently by Sullivan).In particular an algorithmic method is given to decide when a c.g.d.a. has the same homotopy type as its cohomology (the c.g.d.a. is called formal in this case).The chief technique is the construction of a canonically filtered model for a commutative graded differential algebra (over a field of characteristic zero) by perturbing the minimal model for the cohomology algebra. This filtered model is also used to give a simple construction of the Eilenberg-Moore spectral sequence arising from the bar construction. An example is given of a c.g.d.a. whose Eilenberg-Moore sequence collapses, yet which is not formal. |
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