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Chow ring and BP-theory of the extraspecial 2-group of order 32. (English) Zbl 1361.55006

Let \(G := 2_{+}^{1 + 4} = D_8\cdot D_8\) denote the extraspecial 2-group of order 32, which has a central extension \(1 \to {\mathbb Z}/2 \to G \to ({\mathbb Z}/2)^4 \to 1\) (that is key to some of the computations in the paper). Of interest in this paper is the mod-2 Chow ring \(CH^*(G)/2\) of \(G\) (or properly, of its classifying space). For \(p\)-groups of order at most \(p^4\), the mod-\(p\) Chow rings were computed by B. Totaro [Proc. Symp. Pure Math. 67, 248–281 (1999; Zbl 0967.14005)].
The mod-2 cohomology of this extraspecial 2-group \(G\) was computed by D. Quillen [Math. Ann. 194, 197–212 (1971; Zbl 0225.55015)]. The author first determines the Chern subring of the cohomology. Using other work of B. Totaro [Group cohomology and algebraic cycles. Cambridge: Cambridge University Press (2014; Zbl 1386.14088)] on transferred Euler classes, the author determines the multiplicative generators of \(CH^*(G)/2\), most of which come from the Chern subring. However, there are additional nilpotent elements, and these are determined to give a precise description of \(CH^*(G)/2\). This is interesting as an example of a case where the Chow ring has nilpotent elements but the cohomology ring does not. The author further considers the associated Brown-Peterson theory and verifies a conjecture of Totaro by showing that \(CH^*(G)/2 \cong BP^*(G)\otimes_{BP^*}{\mathbb Z}/2\).
Now let \(G\) be an arbitrary group and \(\Omega^*(G)\) denote the BP-version of algebraic cobordism of the classifying space of \(G\). For certain connected algebraic groups and finite groups it is known that the realization map induces an isomorphism between \(\Omega^*(G)\) and \(BP^*(G)\); a fact that is conjectured to hold more generally. Suppose that \(G\) is a \(p\)-group satisfying the mod-\(p\) Totaro conjecture as above (i.e., \(CH^*(G)/p \cong BP^*(G)\otimes_{BP^*}{\mathbb Z}/p\)). In the spirit of earlier arguments, the author outlines a potential inductive argument for verifying this cobordism conjecture for \(G\). Given a subgroup \(M\) of \(G\) with an extension of the form \(1 \to M \to G \to {\mathbb Z}/p^s \to 1\) and such that \(\Omega^*(M) \cong BP^*(M)\), the author identifies a spectral sequence condition that would allow one to conclude that \(\Omega^*(G) \simeq BP^*(G)\). The author is unable to verify that condition for the extraspecial 2-group under consideration earlier, but is able to apply the strategy to several other families of \(p\)-groups.

MSC:

55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
14C15 (Equivariant) Chow groups and rings; motives
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
20J06 Cohomology of groups
55R12 Transfer for fiber spaces and bundles in algebraic topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57R90 Other types of cobordism
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