Induced map on $H_4$ of Eilenberg–MacLane spaces

Question

$\DeclareMathOperator\Hom{Hom}$It is well-known (see Breen, Mikhailov, Touzé - Derived functors of the divided power functors for example) that for $A$ a free abelian group we have $$ H_i(K(A,1); \mathbb{Z}) \cong {\bigwedge}^i A$$

the exterior powers of $A$ and that for $B$ an abelian group we have

$$ H_2(K(B,2); \mathbb{Z}) = B, \quad H_3(K(B,2); \mathbb{Z}) = 0, \quad H_4(K(B,2); \mathbb{Z}) = \Gamma(B)$$

where $\Gamma(B)$ is Whitehead's gamma group. It is the universal recipient of a quadratic map. It can be defined as the free group generated by the symbols $\gamma(b)$ for $b \in B$ subject to two relations:

1. $\gamma(-b) = \gamma(b)$;
2. $\gamma(x + y + z) + \gamma(x) + \gamma(y)+ \gamma(z) = \gamma(x+y) + \gamma(y+z) + \gamma(z+x)$.

A map $f:K(A, 1) \to K(B,2)$ is given by a degree 2 cohomology class. We can use universal coefficients to compute this, and since $A$ is free the ext term vanishes, giving us: $H^2(K(A,1); B) = \Hom( \wedge^2 A, B)$.

Thus given $f: {\bigwedge}^2 A \to B$, we have a map $f:K(A, 1) \to K(B,2)$, and thus an induced map on degree 4 homology: $f_*: {\bigwedge}^4 A \to \Gamma(B)$.

Question: Given $f$, what is this map $f_*: {\bigwedge}^4 A \to \Gamma(B)$?

Another way to phrase this is if I have a quadratic function on $B$ and a homomorphism $f$, then I should be able to combine these into a linear function on ${\bigwedge}^4 A$. How to do this?

Answer 1

Let me summarize comments above. Functoriality of the resulting map $f_*:H_4(K(A,1),\mathbb{Z})\to H_4(K(B,2),\mathbb{Z})$ translates into a request for the universal map $\wedge^4 A\to \Gamma(A)$ which in turn, being functorial in $A$, allows to guess the answer assuming $A$ is free and $rk\ A=4$.

Note that $\Gamma(A)$ identifies with the second divided powers $\Gamma^2(A)$ under the map $\gamma(x)\to \frac{x^2}{2}=\gamma^2(x)$. Thus we can write the product in divided powers as $x\cdot y=\gamma(x+y)-\gamma(x)-\gamma(y)$.

Consider a map given by $$a\wedge b \wedge c \wedge d\to a\wedge b\cdot c\wedge d - a\wedge c\cdot b\wedge d+a\wedge d\cdot b\wedge c$$ Under assumption $rk\ A=4$ it defines a functorial map $\wedge^4 A\to \Gamma(A)$ which is unique up to a scalar. To ensure that the universal $f_*$ is given by the above formula, we have to check that the scalar is indeed equal to $1$. For a prime $p$ we have the reduction inclusion $H_4(K(\wedge^2 A,2),\mathbb{Z})/p\to H_4(K(\wedge^2 A,2),\mathbb{Z}/p)$, thus it is enough to consider similar maps $\bar{f_*}:H_4(K(A,1),\mathbb{Z}/p)\to H_4(K(\wedge^2 A,2),\mathbb{Z}/p)$ for all primes $p$, which are compatible with $f_*$ under the reduction. Then, following the definitions and Serre's description of EM-space cohomology $\mod p$, one can see that its dual is given by the usual multiplication $Sym^2(\wedge^2 (A/p)^*)\overset{\wedge}{\to} \wedge^4 (A/p)^*$. Dualizing again we see that our universal formula is simply the lifting of this comultiplication and there are no unexpected multipliers.