Generalized Euler classes, differential forms and commutative DGAs

TL;DR

This paper explores how iterated odd spherical fibrations in commutative differential graded algebras over rationals produce total spaces with only odd-degree cohomology and demonstrates injectivity of cohomology maps for certain fibrations.

Contribution

It introduces a new construction of total spaces with specific cohomological properties and proves injectivity of cohomology maps in this context, advancing the understanding of DGAs.

Findings

Iterated odd spherical fibrations yield DGAs with only odd degree cohomology.

Cohomology maps induced by fibrations with finite cohomological dimension are injective.

The results connect spherical fibrations with cohomological properties in commutative DGAs.

Abstract

In the context of commutative differential graded algebras over $\mathbb Q$, we show that an iteration of "odd spherical fibration" creates a "total space" commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its "fibrations" with "fiber" of finite cohomological dimension, the induced map on cohomology is injective.