Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the post). In particular, if $n$ is even, it switches $w_2$ and $w_2+w_1^2$. As these classes are the obstructions for existence of $Pin_{+}$ and $Pin_{-}$ structures, … Read more

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

algebraic structure of Integral Steenrod squares

It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations $$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$ In the case where $a$ is odd, one can define an integral refinement $$Sq_{\mathbb{Z}}^a=\beta(Sq^{a-1})\rho_2$$ where $\beta$ is the Bockstein corresponding to the sequence $$\mathbb{Z} \overset{\times 2}{\to} \mathbb{Z} \overset{\rho_2}{\to} \mathbb{Z}/2.$$ This does indeed refine the usual Steenrod squares since … 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