# “a sign that one should be computing K-theory”

Allen Knutson said here in comments below the question that

I generally regard torsion in (co)homology as a sign that one should be computing K-theory instead, which has less of it.

I know one or two things about torsion groups, examples of cohomology groups that has non zero torsion part, some definitions $$K$$-theory.

Can some one help me to understand why torsion in cohomology should remind about computing $$K$$-theory? Any references are welcome.

Some not very well posed question is : what are (some of the) other signs that one should be computing K-theory?

• Thank you @Todd Trimble.. :) – Praphulla Koushik Oct 2 at 11:30

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 object has a complicated structure, then it is a reasonable guess that $$X\simeq Y$$. You can do this kind of argument with ordinary cohomology or complex $$K$$-theory or Morava $$E$$-theory or any of various other invariants; you just choose whichever one seems likely to be effectively calculable and sufficiently rich as to provide the information that you need. A common case is where $$X$$ is equivalent or closely related to a Borel construction $$(Z\times EG)/G$$ for some finite group $$G$$ and some $$G$$-space $$Z$$. In those cases, $$H^*(X)$$ will be related to $$H^*(BG)$$, and $$H^*(BG)$$ is typically an ugly sort of ring with many generators and random-looking relations and lots of torsion. (In fact, if $$|G|=n$$ and $$a\in H^i(BG)$$ with $$i>0$$ then $$na=0$$.) On the other hand, the Atiyah-Segal completion theorem says that $$KU^0(BG)$$ is a completion of the representation ring $$R(G)$$, and $$R(G)$$ is small, nicely structured and free of torsion. So $$K$$-theory is likely to be a more tractable invariant in these cases. The Morava $$E$$-theory ring $$E^0(BG)$$ is often free of torsion as well, and is a richer invariant than $$KU^0(BG)$$.
• The Atiyah-Segal completion theorem provides a clean statement about a forbidding space, $K(\mathbb RP^\infty)=\mathbb Z\oplus\mathbb Z_2$. But this is pretty much equivalent to a crude statement about manifolds: $K(\mathbb RP^{2n})=\mathbb Z\oplus \mathbb Z/2^n$, compared to $H(\mathbb RP^{2n})=\mathbb Z\oplus(\mathbb Z/2)^n$. Both have torsion, indeed groups of the same size, but the number of generators is a measure of "how much," a way in which $K$-theory has "less." – Ben Wieland Oct 3 at 0:47
• Any textbook on topological $K$-theory should cover this example, because it's the first example of $K$-theory being different from homology. The upper bound comes from the Atiyah-Hirzebruch spectral sequence, or computing $K$-theory cell by cell. More interesting is the lower bound, that $K$-theory has torsion of high order. This is a Chern character exercise. The total Chern character $1+c_1+c_2+...$ is a homomorphism from $K$-theory to the multiplicative group of the units of the homology ring. Even though homology is additively 2-torsion, it can detect high torsion in $K$-theory. – Ben Wieland Oct 4 at 16:26