What part of the fundamental group is captured by the second homology group?

Question

Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that $H_1(X)$ is isomorphic to the abelianization of $\pi_1(X)$.
I'm thinking about invariants of 2-knots which can be extracted from have to do with the second homology of (covers of) their complements, and I'm therefore very much interested in the answer to the following question:

What part of the fundamental group is detected by $H_2(X)$?

In particular, is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$?
Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?

Answer 1

$H_2(X)$ is all about $\pi_1(X)$ and $\pi_2(X)$. If $\pi_2(X)$ is trivial (as for knot complements) then it is a functor of $\pi_1(X)$.

Let $H_n(G)$ be $H_n(BG)$, the homology of the classifying space ($K(G,1)$). If $X$ is path-connected than there is a surjection $H_2(X)\to H_2(\pi_1(X))$ whose kernel is a quotient of $\pi_2(X)$, the cokernel of a map from $H_3(\pi_1(X))$ to the largest quotient of $\pi_2(X)$ on which the canonical action of $\pi_1(X)$ becomes trivial.

This $H_2(G)$ isn't anything like the next piece of the derived series after $H_1(G)=G^{ab}$, though. For example, if $G$ is abelian then $H_2(G)$ is the second exterior power of $H_1(G)$ (EDIT: so it can be nontrivial even though it knows no more than $H_1(G)$ does), while if $H_1(G)$ is trivial $H_2(G)$ is often nontrivial (EDIT: so, even when it does carry some more information than $H_1(G)$, it is not necessarily derived-series information).

EDIT: The previous paragraph comes from looking at the integral homology Serre spectral sequence of $X\to K(\pi_1(X),1)$, where the homotopy fiber is the universal cover $\tilde X$. Since $H_1\tilde X=0$, the groups $E^\infty_{p,1}$ are trivial and we get an exact sequence $$ 0\to E^\infty_{0,2}\to H_2(X)\to E^\infty_{2,0}\to 0, $$ therefore $$ E^2_{3,0} \to E^2_{0,2}\to H_2(X)\to E^2_{2,0}\to 0. $$ Since $H_2(\tilde X)=\pi_2(\tilde X)=\pi_2(X)$, this looks like $$ H_3(\pi_1(X)) \to H_0(\pi_1(X);\pi_2(X))\to H_2(X)\to H_2(\pi_1(X))\to 0. $$ The place to look for the rest of the derived series would be homology with nontrivial coefficients, for example homology of covering spaces.

Answer 2

If $X$ is a $K(G,1)$, then the homology is that of the fundamental group, and you'd have Hopf's formula.

If $G = F/R$ where $F$ is a free group, then the formula says that $H_2(G,\mathbb{Z}) \cong (R \cap [F,F])/[F,R]$.

Answer 3

A slight expansion on my comment, sort of complimentary to Tom's response.

In complete generatlity $H_2X$ tells you nothing about $\pi_1 X$.

If $X = A \times B$ with $A$ a $K(\pi,1)$ and $B$ a $K(\pi,2)$, provided $H_2(A)=0$, you have that $H_2 X = H_2 B$.

Since there are lots of $K(\pi,1)$ spaces with $H_2$ trivial, this allows you to construct many spaces with identical fundamental groups yet $H_2$ varies wildly.

You'll want to restrict to fairly particular spaces to avoid this independence.

edit: If you're happy taking covering spaces then $H_2$ (of of an arbitrary cover of $X$) starts to see quite a bit more of $\pi_1 X$. If $\widetilde{X} \to X$ is the universal cover then $H_2 \widetilde{X} \simeq \pi_2 X$ by the Hurewicz theorem. So now Tom's comments apply, giving you a concrete relationship between $H_2 X$, $\pi_1 X$ and $\pi_2 X = H_2 \widetilde{X}$.

Answer 4

For a group $G$, $H_2(G,\mathbb{Z})$ is also called the Schur multiplier of $G$. Among other things, if $G$ is perfect (ie $H_1=0$) then it is a term in the universal central extension $\widehat{G}$for $G$. That is, you have a short exact sequence

$1\to H_2(G,\mathbb{Z})\to\widehat{G}\to G\to 1$ .

(The wikipedia article focuses on the case of $G$ finite, but this works in greater generality.)

Answer 5

thinking about invariants of 2-knots ... is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$? Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?

Have you heard of things like Dwyer's filtration on $H_2$ and Stallings/Cochran-Harvey theorems about the lower central/derived series? I cannot resist quoting the opening sentence of Krushkal's paper:

"The lower central series of the fundamental group of a space $X$ is closely related to the Dwyer’s [D] filtration $\phi_k(X)$ of the second homology $H_2(X;\Bbb Z)$."

The Dwyer subspace $\phi_k(X)\subset H_2(X;\Bbb Z)$ is defined as the kernel of the composition $$H_2(X) \to H_2(\pi_1X) \to H_2(\pi_1X/\gamma_{k-1}\pi_1X).$$ The gamma notation for the lower central series runs $\gamma_1G=G$, and $\gamma_{k+1}G=[G,\gamma_k G]$.

Freedman and Teichner showed that

$\phi_k(X)$ coincides with the set of homology classes represented by maps of closed $k$-gropes into $X$.

Now their $k$-gropes are gropes of class $k$, but you can also do symmetric gropes of depth $k$, or whatever they are called, to get into the derived series. Which series to choose, and how to make use of it? The lower central series has a long standing record of applications in knot/link theory due in part to Stallings' theorem (1963, also rediscovered by Casson; note Cochran's topological proof):

Let $\phi:A\to B$ be a homomorphism that induces an isomorphism on $H_1(−;\Bbb Z)$ and an epimorphism on $H_2(−;\Bbb Z)$. Then, for each $n$, $\phi$ induces an isomorphism $A/\gamma_nA\simeq B/\gamma_nB$.

Dwyer (1975) extended Stallings’ theorem by weakening the hypothesis on $H_2$:

Let $\phi: A\to B$ be a homomorphism that induces an isomorphism on $H_1(−;\Bbb Z)$. Then for any $n\ge 2$ the following are equivalent:

• $\phi$ induces an isomorphism $A/\gamma_nA\simeq B/\gamma_nB$;

• $\phi$ induces an epimorphism $H_2(A;\Bbb Z)/\phi_n(A)\simeq H_2(B;\Bbb Z)/\phi_n(B)$.

Now if you do want the derived series rather than the lower central series, take a look at the work of Cochran and Harvey who had a series of papers about the analogues of the Stallings and Dwyer theorems for torsion-free derived series. In fact their motivation was also knot theory.

Also Mikhailov has some other generalization of the Dwyer filtration (see also his book with Passi in Springer Lecture Notes in Math) though I'm not sure if this has had any application to knots.

In fact, I think Mikhailov, and possibly Orr and Cochran, also did something about the transfinite Dwyer filtration, which might be not unrelevant to knots and links (at least this is where the transfinite business originated from, in papers by Orr and Levine in the 80s).

Answer 6

There is another description of $H_2(G)$ due to Miller:

Miller, Clair The second homology group of a group; relations among commutators. Proc. Amer. Math. Soc. 3, (1952). 588--595.

In modern terminology, let $x,y\in G$ and denote ${}^xy=xyx^{-1}$. Let $G\wedge G$ be the group generated by the symbols $x\wedge y$, $x,y\in G$, subject to the following relations:

$$ xy\wedge z = ({}^xy\wedge {}^xz)(x\wedge z),\quad x \wedge yz = (x\wedge y)({}^yx\wedge {}^yz),\quad x\wedge x = 1, $$ for all $x,y,z\in G$. The group $G\wedge G$ is also known as the nonabelian exterior square of $G$. There is a commutator homomorphism $\kappa :G\wedge G\to [G,G]$ given by $x\wedge y\mapsto [x,y]$. Miller essentially proved that $\ker \kappa$ is naturally isomorphic to $H_2(G)$.

This is closely related to the answers of Henry Wilton and Richard Kent. Namely, $G\wedge G$ is isomorphic to the derived subgroup of a covering group $\hat{G}$ of $G$, and it is not difficult to find an isomorphism between $\ker\kappa$ and $(R\cap [F,F])/[F,R]$.

Answer 7

Hopf's formula gives a determination of $H_2(G)$ in terms of a presentation of the group $G$. Higher Hopf type formulae for $H_n(G)$ were given in terms of double and multiple presentations of $G$ in

R. Brown and G.J. Ellis, ``Hopf formulae for the higher homology of a group'', Bull. London Math. Soc. 20 (1988) 124-128.

There are now a number of papers in this area as is shown by the references in for exanple

Everaert, Tomas; Gran, Marino; Van der Linden, Tim: Higher Hopf formulae for homology via Galois theory. Adv. Math. 217 (2008), no. 5, 2231–2267.

Miller's result is retrieved in

R. Brown and J.-L. Loday ``Van Kampen theorems for diagrams of spaces'', Topology 26 (1987) 311-334.

A tensor product $G \otimes G$ is introduced so that the commutator map determines a morphism $\kappa: G \otimes G \to G$ whose image is the commutator subgroup. It is proved that Ker $\kappa$ is isomorphic to $\pi_3 S K(G,1)$. In fact the 3-type of $SK(G,1)$ is completely determined. Item 114 of the bibliography Tim mentions links to computer calculations of this tensor product.

The nice point is that $\kappa:G \otimes G \to G$ has, like other tensor products in mathematics, a universal property, and this is not shared by the commutator sugroup of $G$!

Ronnie Brown

Answer 8

Maybe it is worth to point out something that is at the origin of the results mentioned in several answers to the question. Hopf in one of his famous papers (Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1940), 257-309) gives a characterization of the cokernel of the Hurewicz map in degree 2. His result states that for an arcwise-connected locally finite simplicial complex $X$, the Hurewicz map $h_2:\pi_2(X)\to H_2(X)$ has cokernel: $$ H_2(X)/h_2(\pi_2(X))\simeq (R\cap [F,F])/[F,R], $$ where $\pi_1(X)\simeq F/R$ is any presentation of the fundamental group.

In the same paper, he observes first that for any finitely generated group $G$ with a presentation $F/R$ the abelian group: $$ G^*_1=(R\cap [F,F])/[F,R] $$ only depends on $G$, and not on the particular presentation $F/R$.

In our modern day language, we would write Hopf's result as stating that: $$ H_2(X)/h_2(\pi_2(X))\simeq H_2(\pi_1(X)), $$ which is the form that appears in some of the answers above.