HotCoalgebras · Homotopy theory of cosimplicial unstable (co-)algebras over the Steenrod algebra
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2015-05-01 → 2017-04-30
- EU contribution
- €185,076
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Homotopy theory of cosimplicial unstable (co-)algebras over the Steenrod algebra
In algebraic topology one studies algebraic invariants of geometric objects up to continuous deformation. The latter is what is called a "homotopy" giving the subject its name: Homotopy Theory. Homotopy theorists engage in defining, interpreting, and computing such invariants. Since its "invention" around 1900 the importance of the field within mathematics has steadily grown. Very recent developments extend its reach to physics and computer science. One important invariant is singular homology. It yields a functor (ie. a natural construction) from topological spaces (ie. geometric objects) to the category of unstable coalgebras over the Steenrod algebra. The general theory of such homotopy-invariant functors is now called Goodwillie calculus. The original proposal laid out the importance of deepening our understanding of the homotopy theory of cosimplicial unstable (co-)algebras over the Steenrod algebra and its relation to the homotopy theory of cosimplicial spaces. It also mentioned the desirability of bringing methods from Goodwillie calculus to bear on this question.
Data: CORDIS, © European Union
Project objective
This research proposal is in mathematics, its content is part of algebraic topology and homotopy theory. It aims at deepening our understanding of the homotopy theory of cosimplicial unstable (co-)algebras over the Steenrod algebra and its relation to the homotopy theory of cosimplicial spaces. This is achieved by new methods developed recently by the ER (Dr. Biedermann) and coauthors and by methods from Goodwillie calculus. Specifically, there are three closely related parts/work packages: 1. Prove a general vanishing theorem of higher obstructions for realizing a map on homology as a map of spaces. The theorem is known to hold in rational homotopy and in the mod p Massey-Peterson case. 2. Find an algebraic description of the first obstruction living in Andre-Quillen cohomology (AQC) to the existence of a realization of unstable coalgebras. 3. Define natural operations on AQC of unstable coalgebras with general coefficients.As part of the risk management we describe two further fallback projects: 4. Study the Goodwillie tower of the identity functor of simplicial unstable algebras and relate its layers to AQC. 5. Describe the algebra of homotopy operations on simplicial commutative algebras for odd primes p. These projects are parts of a program of the ER to investigate realization problems and rigidity results associated to singular (co-)homology. A longterm goal (beyond the time frame of the fellowship) is to develop a deformation theory of unstable (co-)algebras over the Steenrod algebra and their realizing homotopy types in the mod p case.
Original text from CORDIS.
Participants
- UNIVERSITE PARIS 13 · VilletaneuseCoordinatorFrance
Links
Data: CORDIS, © European Union