Brown–Peterson cohomology and Morava K-theory of DI(4) and its classifying space
نویسنده
چکیده
DI(4) is the only known example of an exotic 2-compact group, and is conjectured to be the only one. In this work, we study generalized cohomology theories for DI(4) and its classifying space. Specifically, we compute the Morava K-theories, and the P (n)-cohomology of DI(4). We use the non-commutativity of the spectrum P (n) at p = 2 to prove the non-homotopy nilpotency ofDI(4). Concerning the classifying space, we prove that the BP -cohomology and the Morava K-theories of BDI(4) are all concentrated in even degrees.
منابع مشابه
Morava Hopf algebras and spaces K(n) equivalent to finite Postnikov systems
We have three somewhat independent sets of results. Our first results are a mixed blessing. We show that Morava K-theories don’t see k-invariants for homotopy commutative H-spaces which are finite Postnikov systems, i.e. for those with only a finite number of homotopy groups. Since k-invariants are what holds the space together, this suggests that Morava K-theories will not be of much use aroun...
متن کاملAn Unstable Change of Rings for Morava E-theory
The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology theory BP, Johnson-Wilson theory E(n), or Morava E-theory En. For nice spaces the E2-term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between ExtUBP∗BP (BP∗,M) and ExtUEn∗En (En∗, En∗⊗BP∗ M) ...
متن کاملBP Operations and Morava's Extraordinary K-Theories
In a series of papers [17-19] Morava uses an infinite sequence of extraordinary K-theories to give an elegant structure theorem for the complex cobordism of a finite complex. Much of Morava's theory is embedded in a rather sophisticated algebraic setting. In our attempt to understand his work, we have found more conventional algebraic topological proofs of many of his results. Also, our approac...
متن کاملThe Cohomology of the Height Four Morava Stabilizer Group at Large Primes
This is an announcement of some new computational methods in stable homotopy theory, in particular, methods for using the cohomology of small-height Morava stabilizer groups to compute the cohomology of large-height Morava stabilizer groups. As an application, the cohomology of the height four Morava stabilizer group is computed at large primes (its rank turns out to be 3440). Consequently we a...
متن کاملA lower bound for coherences on the Brown–Peterson spectrum
We provide a lower bound for the coherence of the homotopy commutativity of the Brown–Peterson spectrum, BP , at a given prime p and prove that it is at least (2p2 + 2p− 2)–homotopy commutative. We give a proof based on Dyer–Lashof operations that BP cannot be a Thom spectrum associated to n–fold loop maps to BSF for n = 4 at 2 and n = 2p + 4 at odd primes. Other examples where we obtain estima...
متن کامل