Timeline for What does Robert Stong mean when he says $H^*(MO(k))$ is a free Steenrod algebra in dimension less than $2k$?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 2 at 8:49 | vote | accept | Chris | ||
| S Dec 2 at 8:35 | history | edited | Achim Krause | CC BY-SA 4.0 |
added 1030 characters in body
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| Dec 2 at 8:19 | comment | added | Chris | Can you edit your answer to explicitly spell this out? If so I will gladly accept you answer. My apologies, I am just trying to fully wrap my head around this because it all feels mildly sketchy. | |
| Dec 2 at 8:17 | comment | added | Achim Krause | Ah. Then he probably deduces the unstable statement from the stable one? Once you know that $H^*(MO)$ is free (which you can prove less explicitly using the Milnor-Moore theorem), the unstable statement follows simply because the cohomology of $MO(r)$ and $MO$ agree in the necessary range of degrees. | |
| Dec 2 at 8:06 | comment | added | Chris | I am more so asking why the statement "The free module structure follows from the stability $\widetilde{H}^{r+i}(MO(r))\cong \widetilde{H}^{r+i+1}(MO(r+1))$ for $i\leq r$." implies that $H^*(MO(r))$ looks like a free module in degrees $\leq 2r$. | |
| Dec 2 at 8:03 | comment | added | Achim Krause | Writing down such a homomorphism just corresponds to choosing a family of elements of degree $n_i$ in the target. Checking that it is an isomorphism in degrees $\leq 2r$ amounts to checking that certain terms are linearly independent and generate. This of course involves serious computations, the details should be explained by Stong. | |
| Dec 2 at 7:39 | comment | added | Chris | Could you potentially elaborate a bit more on how we justify this isomorphism? For example, Stong seems to presuppose that we can get maps like this and uses it to get a homotopy equivalence to Eilenberg McLane spaces, so I am not sure how to justify that such an isomorphism exists. | |
| Dec 2 at 7:37 | review | Suggested edits | |||
| S Dec 2 at 8:35 | |||||
| Dec 2 at 7:31 | history | answered | Achim Krause | CC BY-SA 4.0 |