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For finite-type torsion spectra $X$ there is a natural isomorphism $$ KU_{n-1}(X) \simeq \text{Hom}_c(KU^n(X),\mathbb{Q}/\mathbb{Z}). $$ Here $\text{Hom}_c$ denotes the group of homomorphisms that ar …

Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles on $X$; in parti …

First, I do not think that strict associativity makes a difference, so I will ignore it. Next, let $G$ be a finite group, and let $\mathcal{C}G$ be the category of finite $G$-sets and equivariant bij …

There is an evident map $\mathbb{R}P^{m-1}\times\mathbb{R}P^{n-1}\to PX$, which is easily seen to be an isomorphism. (This is called the Segre embedding.) If we pull back the tautological bundle of …

The most explicit answers are in work of Sarah Whitehouse and her collaborators. You could start with this paper and its references: http://www.ams.org/journals/proc/2010-138-06/S0002-9939-10-10237- …

I'll just give an answer for finite complexes $X$. The condition $\widetilde{K}^*(X)=0$ only depends on the suspension spectrum of $X$ so this is naturally regarded as a question in stable homotopy th …

Here are some comments about the case where $X$ is not assumed to have finite type. Put $Y=\Sigma X$. For any field $K$, the groups $H_*(Y;K)$ form a Hopf algebra in which all elements of the augmen …

You have given yourself an invertible element $u\in\pi_2(R)$ and a coordinate $x\in R^2(\mathbb{C}P^\infty)$ with $\psi(x)=x\otimes 1 + 1\otimes x + ux\otimes x$. This means that the class $m=1+ux\in …

The cobordism groups can be calculated using the Adams spectral sequence, which is based on the homological algebra of modules over the Steenrod algebra. This works most nicely for the unoriented cob …

There are cofibrations $$\Sigma^2 kU\xrightarrow{v} kU\to H\xrightarrow{\alpha}\Sigma^3 kU$$ and $$ H\xrightarrow{p}H\to H/p\xrightarrow{\beta} \Sigma H. $$ The composite $\alpha\beta\colon H/p\to\ …

Adams operations exist in quite wide generality. For any even periodic ring spectra $E$ and $F$, we have associated formal groups $G_E$ and $G_F$ over base schemes $S_E$ and $S_F$. There is a moduli …

Depending on exactly what you are trying to do, you may find it more useful to consider $E^0(BV)$, where $E$ is Morava $E$-theory and $V\simeq(\mathbb{Z}/p)^d$. (Here everything is $2$-periodic and co …

This seems to be an answer, based on discussion with Maxime Ramzi in the comments. The space of $E_\infty$ $KU$-algebra maps from $KU\otimes S^1_+\to F(S^1_+,KU)$ is the same as the space of $E_\infty …

Usually you are computing $H^*(X)$ or $K^*(X)$ for a reason; for example if $H^*(X)\not\simeq H^*(Y)$ then you know that $X$ and $Y$ are not homotopy equivalent, but if $H^*(X)\simeq H^*(Y)$ and this …

First, for any bundle $V$ of dimension $d$ over $Y$ put $$ \lambda(V)(t) = \sum (-1)^k[\Lambda^k(V)]t^{d-k} \in K^0(Y)[t]. $$ This is a monic polynomial of degree $d$ over $K^0(Y)$. It satisfies $\l …