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I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks …

Let $G$ be a topological group and $X$ be a paracompact Hausdorff topological space. For simplicity let us assume that $G$ has the homotopy type of a CW complex, although a lot of this answer does not …

It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although most of them …

I apologize for the self promotion -- I hope the content of this answer can be useful anyway... My favourite introduction to L-theory is Lurie's notes on Algebraic L-theory and surgery (warning: aggr …

Let me first write what happens for classical cobordism. You are basically asking whether the map $\operatorname{MU}\to H\mathbb{Z}$ factors through the projection $\operatorname{ku}\to H\mathbb{Z}$. …

My favourite reference for understanding spectral sequences is Boardman, J. Michael. "Conditionally convergent spectral sequences." Contemporary Mathematics 239 (1999): 49-84 (pdf). I don't think I …

Let $X$ be a finite complex. Then the functor $$\lim_X:\operatorname{Fun}(X,\operatorname{Sp})\to \operatorname{Sp}$$ sending a local system of spectra $E$ to its limit preserves all colimits. Indeed …

I think thinking in terms of classifying spaces will help clarify the situation. We know that a rank $n$ complex vector bundle $V$ on $X$ is the same thing as a homotopy class of maps $f:X\to BU_n$. A …

Since the question remains unanswered, let me copy Tom Goodwillie's comment: If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely g …

As a first introduction I like these notes by Achim Krause and Thomas Nikolaus. They do require some familiarity with spectra and stable homotopy theory though.

(This answer is written in a model-independent fashion -- translate to your favourite formalism). For every path $\gamma:[0,1]\to B$ you get an isomorphism in the homotopy category $X_{\gamma0}\xrigh …

These are known as acyclic spaces (note that since $\tilde C_*(X)$ is a bounded below complex of projectives, it being contractible is equivalent to its homology being trivial). There's an extensive …

What you're missing is that $[\mathbb{S},\mathbb{S}]=\mathbb{Z}$. Let now $R$ be the infinite matrix of integers representing $\rho$. Note that since $\rho$ takes value in $\bigoplus_{I_1}\mathbb{Z}\s …

This is an assemblage of known results, I'll try to put a reference for all of them. By a classical theorem of Serre all stable homotopy groups are finite in positive degree. In particular we have $ …

I think you made a sign mistake, and asked for a left adjoint to $\Omega^\infty$ since the monoid ring is a left along to the forgetful functor. If so then the answer is yes. $\Sigma^\infty_+:\mathr …