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I apologize that this is perhaps not adequate for mathoverflow but I have struggled with this for days now and become desperate... The reduced K-group $\tilde{K}(S^0)$ of the zero sphere is the ring …

Fix a prime $p$ and consider everything mod $p$. Steenrod operations arise somehow from the loss of information passing from the singular complex of a space to its cohomology ring. Are they exactly th …

A spectrum is a sequence $X_0,X_1,...$ of spaces together with structure morphisms $\Sigma X_n\to X_{n+1}$. To get the usual model for the stable homotopy category based on the category of spectra, on …

Which of the statements is wrong: a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point reduced complex $K$-theory $\tilde K$ and reduced real $ …

(I apologize that this is a vague question). I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups fr …

In the top answer to the question "Is there a map of spectra implementing the Thom isomorphism?" it is explained (with a reference to Rudyaks book) that from a rank $r$ vector bundle $\mu:V\to X$, a s …

Define a reflexive relation on the set of zero-simplices of a simplicial set $A$ by saying that $x\sim y$ iff there is a one-simplex $h$ with $0$-face $y$ and $1$-face $x$. This is not an equivalence …

Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\ma …

It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal). cofibrations and weak eq …

(This is a follow-up question to this one). As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well …

What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. Yo …

Let $X$ be a finite CW complex. Swan's theorem provide an equivalence $$ {\rm Vec}(X)\xrightarrow\sim{\rm ProjMod}(\mathop{\rm hom}\nolimits_{\rm Top}(X,\mathbb{R})) $$ between the category of finite …