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[Edit: in response to prof. Sullivan's answer, let me try to expand the question a bit.]

Let $G$ be a discrete group which is perfect (trivial abelianization) and let $BG \to BG^+$ denote Quillen's plus construction, i.e., an acyclic map to a simply connected space. $BG$ comes with a preferred basedpoint and I'll use the image in $BG^+$ as basepoint there. Stable homotopy of $BG^+$ is defined as $\pi_k^s(BG^+) = \mathrm{colim}_n \pi_{n+k} (S^n BG^+)$ and comes with a "Hurewicz map" $\pi_k(BG^+) \to \pi_k^s(BG^+)$. One easily shows

Lemma: If $BG^+$ is an infinite loop space then this Hurewicz map $\pi_k(BG^+) \to \pi_k^s(BG^+)$ is split injective.

The idea of the proof is that if $BG^+ = \Omega^\infty X$ for a spectrum $X$, then the map $(\Omega^\infty X) \hookrightarrow \Omega^\infty S^\infty (\Omega^\infty X)$ adjoint to the identity of $S^\infty \Omega^\infty X$ is split by $\Omega^\infty \mathrm{ev}: \Omega^\infty (S^\infty \Omega^\infty X) \to \Omega^\infty X$.

Now the fact that $BG \to BG^+$ is acyclic implies that it induces an isomorphism $\pi_k^s(BG) \to \pi_k^s(BG^+)$, so $\pi_k(BG^+)$ is identified with a summand of $\pi_k^s(BG)$. On the other hand, we may use Pontryagin-Thom to identify $\pi^s_k(BG)$ with pairs $(M,f)$ where $M$ is a stably framed compact $k$-manifold and $f: M/\partial M \to BG$ is a pointed map, up to cobordism of such pairs.

Question: Is there a simple criterion for when an element of $\pi_k^s(BG)$ is in the image of $\pi_k(BG^+) \to \pi_k^s(BG^+) = \pi_k^s(BG)$?

Now "simple criterion" is of course not very precise, so let me give an example. It is easy to see that (for $k \geq 2$) it is sufficient to be representable by $(M,f)$ with $M/\partial M$ a homology sphere. Indeed, we can plus-construct the map $f: M/\partial M \to BG$ to get $S^k \simeq (M/\partial M)^+ \to BG^+$. Is this simple and sufficient criterion also necessary? If not, is there some other "nice" description of this summand $\pi_k(BG^+) \subset \pi_k^s(BG^+) = \pi_k^s(BG)$?

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I do not think the question is completely formulated. However , one may note, there is a coincidence of notation between the initial part with BG for a perfect group G and the second part which refers indirectly to the well studied in geometric topology infinite loop space BG whose homotopy groups are the stable homotopy groups of spheres. This coincidence** reminds me of some ancient facts and unrecorded history* about the plus construction which unifies the two interpretations of the symbol BG in the two paragraphs of the question.

Firstly, one may apply the attaching two and three cells construction , marvelously used and named "plus construction" by Dan Quillen to define and compute the algebraic K groups of a ring R and later replaced by the group completion in this setting, can be applied to any space not just spaces with perfect fundamental group and contractible universal covering spaces. For example, applying the "two and three cell" construction that preserves homological type [my name for the later named plus construction] to the infinite increasing union of the finite symmetric groups which has Z/2 as its abelianization one obtains the infinite loop space BG in its second interpretation. This beautiful theorem of Priddy is referenced and discussed in "Geometric Topology: Localization , Periodicity and Galois symmetry MIT Notes 1970 in { K-theory Monographs Kluwer Press by this author }

The two and three cell construction or [2,3]-construction which kills any perfect subgroup of the fundamental group and preserves homology came up in a conversation with Dan Quillen at the IAS in 67 or 68 when Dan showed me his calculations of the cohomology at any prime l not equal to p of BG for G equal to Gl(n) over the prime field at p. This gave a beautiful polynomial algebra on chern classes up to n. It struck me that this calculation gave a new way to see the complex Adams conjecture, concerning the structure of the natural map in homotopy theory of BU into the BG just mentioned above , which classified stable sphere bundles up to fibrewise homotopy equivalence. This, because applying the [2,3]-construction to his BGL(n) for the prime field at p gave a model of BGL(n,C) localized at any l not equal to p. We discussed that combining this model with his much earlier idea that the Frobenius at p was a formal analogue at the prime l of the pth Adams operation which is a homotopy equivalence at any prime l not equal to p, AND with my finding from earlier in 67 that the Adams conjecture for the p-th Adams operation as a self mapping of BU was equivalent to being able to lift the Adams operation back to a large unstable skeleton of BU(n) [ all of this explained in the reference above]. Both of us were too busy to write up something from that conversation but continued developing respective lines of research in progress, including our respective proofs of the A dams conjectures.

  • Although it seems Dan Quillen told something about this story at his 60th fest. ** I trust these connections will be interesting to some and may stimulate the formulation of a question I can understand better.

Dennis Sullivan ,Sunday December 15th 2019.

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Here is an attempt, under the assumption that the universal central extension of $G$ admits a finite presentation. Instead of studying relative framed bordism of the pair $(BG,\ast)$ it studies the pair $(BG,F)$, where $F \to BG$ is the homotopy fiber of $BG \to BG^+$. The canonical map $\pi_\ast^s(BG,F) \to \pi_\ast^s(BG,\ast)$ is an isomorphism, but it seems easier to formulate a criterion in terms of framed bordism representatives of the pair $(BG,F)$.

Preliminary observations: Write $\widetilde{G} \to G$ for the universal central extension and build a 2-dimensional CW complex $F'$ with one 0-cell $x$, a 1-cell for each generator in the presentation, and a 2-cell for each relation, such that $\pi_1(F',x) \cong \widetilde{G}$. The Hurewicz map $\pi_2(F',x) \to H_2(F';\mathbb{Z})$ is surjective and its codomain is a free abelian group of finite rank, so we choose a basis represent each basis vector by a map $S^2 \to F'$. Let $F$ be the 3-dimensional CW complex obtained by attaching a cell along each of these maps. Then $F$ is acyclic and comes with an isomorphism $\pi_1(F,x) \cong \widetilde{G}$. There is a unique homotopy class of based maps $$F \to BG$$ inducing the canonical homomorphism $\widetilde{G} \to G$ on fundamental groups. It is easy to verify that the homotopy cofiber of this map is a model for $BG^+$ and that the map itself is a model for the homotopy fiber of $BG \to BG^+$. We deduce in particular that this homotopy fiber has the homotopy fiber of a finite 3-dimensional CW complex. (I learned this from Kervaire's paper "Smooth homology spheres and their fundamental groups".)

Let $\ast \in BG^+$ denote the basepoint and let $f: (D^n,\partial D^n) \to (BG^+,\ast)$ be a map, representing $[f] \in \pi_n(BG^+,\ast)$. Replace $BG \to BG^+$ with a Serre fibration and write $Y \to D^n$ for the pullback along $f$ and $X \to \partial D^n$ for the restriction of that. This pair of spaces comes with a canonical map $$g: (Y,X) \to (D^n,\partial D^n)$$ which induces an isomorphism in relative integral homology. There are weak equivalences $Y \simeq F$ and $X \simeq \partial D^n \times F$, and in particular both spaces have the homotopy type of a finite complex.

The map $g$ also induces an isomorphism in framed bordism, so there exists a framed compact $n$-manifold $M$ and a continuous $(M,\partial M) \to (Y,X)$ whose composition with $g$ is framed bordant to the identity map of $(D^n,\partial D^n)$. I think the pair $(Y,X)$ is a "Poincaré pair" and that Wall's $\pi$-$\pi$-theorem implies that there exists a representative in which $(M,\partial M) \to (Y,X)$ is a weak equivalence, provided $n \geq 6$. This implies that $M/\partial M \simeq Y/X \simeq S^n$.

It seems we have proved that (at least for $n \geq 6$ and under the finiteness assumption) an element of $\pi_n^s(BG,F)$ is in the image of the Hurewicz map $\pi_n(BG^+,\ast) \to \pi_n^s(BG^+,\ast) \cong \pi_n^s(BG,F)$ if and only if it admits a representative $(M,\partial M) \to (BG,F)$ in framed bordism, where $M/\partial M$ is a homotopy sphere.

If we try the same strategy with the pair $(BG,\ast)$ instead of $(BG,F)$, we won't have the $\pi$-$\pi$-theorem available. Instead we will have obstructions in the relative $L$-groups of $\mathbb{Z} \to \mathbb{Z}[G]$. It seems possible that the sufficient criterion in the question is not necessary if these relative $L$-groups don't vanish, but at the moment I don't have a counterexample either.

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