Generation of cohomology of graded algebras

Let A be an unital, associative, graded algebra over a base ring k. I’m happy to assume that k is a field if need be, and will insist that A free and of finite rank in each degree (locally finite). Further, A is connected: it vanishes in negative degrees and is of rank 1 (generated … Read more

Adams Spectral sequence and Pontrjagin-Thom construction [Reference request]

I will be grateful for any reference for the following statements/claims. 1) Let’s consider the case of $p=2$ and the classic Adams spectral sequence with the $E_2$-term given by $\mathrm{Ext}_{A}(\mathbb{F}_2,\mathbb{F}_2)$. If $\alpha$ and $\beta$ are two permanent cycles in the Adams spectral sequence, converging to elements $f\in{_2\pi_i^s}$ and $g\in{_2\pi_j^s}$, then is it true that $\alpha\beta$ … Read more

Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let X be a scheme of finite type over C and let Z↪X be a closed subscheme. Consider the associated closed inclusion Zan↪Xan between their analytifications (regarded as topological spaces). Is this a strong neighborhood deformation retract? By this I mean, can I find for every neighborhood U of a point z in Zan another … Read more

Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices. The (conjugate) Steinberg idempotent is defined to be $$e_n’=\frac{1}{q_n}\Sigma _{g\in \Sigma _n}sgn(g)g \Sigma _{g\in B _n}g \in Z_{(2)}[Gl_n(Z/2)]$$ where $q_n$ is an appropriate odd number, $sgn$ is the signature of the permutation. Now, let’s … Read more

Milnor’s model of EGEG and Kac-Moody groups

I am working with non-compact Kac-Moody groups K. We can use Milnor’s join model for EK=lim→K∗n, where K∗n is the iterated join (see page 20 of this PDF). Let KJ be a parabolic subgroup of K. I would like to use the system {K∗n} as a model for EKJ and compare it to the system … Read more

Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I’m a novice in the algebraic topology, so I’m unable to resolve it by myself. Definition Let us call a (pure) n-dimensional polyhedral complex the topological space glued from a finite number of n-dimensional (convex) polyhedra … Read more

References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms ⟨⋅,⋅⟩i:Ci×Ci→K defined for all i on chain complexes ⋯→Ci+1→Ci→Ci−1→⋯ of K-modules over a principal ideal domain K. I am particularly interested in how these descend to homology groups and what invariants/classifications can be … Read more

Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if A=C∗(X;Q) is the CDGA of rational cochains on X then the cohomology of the bar complex of A is isomorphic to H∗(ΩX;Q) as a coalgebra (see e.g. Félix–Oprea–Tanré, Algebraic models in geometry, Theorem 5.52 for this formulation). Maybe this is a naive question, but if I take a rational … Read more

A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space G/K admits a homomorphism from H∗(BK) induced from the classifying map G/K→BK of the principal K-bundle G→G/K. Assume the Lie group is K connected, so that π1(BK)=0; then the space G/K is formal in the sense of rational homotopy theory if and only if H∗(G/K) is a free … Read more