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There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion $$ BGL_1(S^0) \to BGL_\infty(S^0) $$ where $BGL_1(S^0)$ is $BG$. Here $BGL_\infty(S^0)$ can be defined as the homotopy colimit over $k$ and $n$ of the topological monoid $G(\vee_k S^n)$ of homotopy automorphisms of a $k$-fold wedge of $n$-spheres.

Question 1: I've heard it mentioned that there can be no retraction $A(*) \to BG$ to the map $BG \to A(\ast)$. Can someone please explain to me why there is no such splitting, and if possible, give a reference?

More generally,

Question 2: If $R$ is a structured ring spectrum, is there a reasonable set of conditions which guarantees that the map $BGL_1(R)\to K(R)$ admits a retraction?

(A related question is under what conditions does an $R$-module map $f: R^n \to R^n$ which is a weak homotopy equivalence admit a determinant $\text{det}(f) \in GL_1(R)$).

There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion $$ BGL_1(S^0) \to BGL_\infty(S^0) $$ where $BGL_1(S^0)$ is $BG$. Here $BGL_\infty(S^0)$ can be defined as the homotopy colimit over $k$ and $n$ of $BG(\vee_k S^n)$, where $G(\vee_k S^n)$ is the topological monoid of homotopy automorphisms of a $k$-fold wedge of $n$-spheres. (Note: $A(\ast) = \Bbb Z \times BGL_\infty(S^0)^+$.)

Question 1: I've heard it mentioned that there can be no retraction $A(*) \to BG$ to the map $BG \to A(\ast)$. Can someone please explain to me why there is no such splitting, and if possible, give a reference?

More generally,

Question 2: If $R$ is a structured ring spectrum, is there a reasonable set of conditions which guarantees that the map $BGL_1(R)\to K(R)$ admits a retraction?

(A related question is under what conditions does an $R$-module map $f: R^n \to R^n$ which is a weak homotopy equivalence admit a determinant $\text{det}(f) \in GL_1(R)$).

There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion $$ BGL_1(S^0) \to BGL_\infty(S^0) $$ where $BGL_1(S^0)$ is $BG$. Here $BGL_\infty(S^0)$ can be defined as the homotopy colimit over $k$ and $n$ of the topological monoid $G(\vee_k S^n)$ of homotopy automorphisms of a $k$-fold wedge of $n$-spheres.

Question 1: I've heard it mentioned that can be no retraction $A(*) \to BG$ to the map $BG \to A(\ast)$. Can someone please explain to me why there is no such splitting, and if possible, give a reference?

More generally,

Question 2: If $R$ is a structured ring spectrum, is there a reasonable set of conditions which guarantees that the map $BGL_1(R)\to K(R)$ admits a retraction?

(A related question is under what conditions does an $R$-module map $f: R^n \to R^n$ which is a weak homotopy equivalence admit a determinant $\text{det}(f) \in GL_1(R)$).

There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion $$ BGL_1(S^0) \to BGL_\infty(S^0) $$ where $BGL_1(S^0)$ is $BG$. Here $BGL_\infty(S^0)$ can be defined as the homotopy colimit over $k$ and $n$ of the topological monoid $G(\vee_k S^n)$ of homotopy automorphisms of a $k$-fold wedge of $n$-spheres.

Question 1: I've heard it mentioned that there can be no retraction $A(*) \to BG$ to the map $BG \to A(\ast)$. Can someone please explain to me why there is no such splitting, and if possible, give a reference?

More generally,

Question 2: If $R$ is a structured ring spectrum, is there a reasonable set of conditions which guarantees that the map $BGL_1(R)\to K(R)$ admits a retraction?

(A related question is under what conditions does an $R$-module map $f: R^n \to R^n$ which is a weak homotopy equivalence admit a determinant $\text{det}(f) \in GL_1(R)$).

There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion $$ BGL_1(S^0) \to BGL_\infty(S^0) $$ where $BGL_1(S^0)$ is $BG$. Here $BGL_\infty(S^0)$ can be defined as the homotopy colimit over $k$ and $n$ of the topological monoid $G(\vee_k S^n)$ of homotopy automorphisms of a $k$-fold wedge of $n$-spheres.

Question 1: I've heard it mentioned that can be no retraction $A(*) \to BG$ to the map $BG \to A(\ast)$. Can someone please explain to me why there is no such splitting, and if possible, give a reference?

More generally,

Question 2: If $R$ is a structured ring spectrum, is there a reasonable set of conditions which guarantees that the map $BGL_1(R)\to K(R)$ admits a retraction?

(A related question is under what conditions does an $R$-module map of $f: R^n \to R^n$ which is a weak homotopy equivalence admit a determinant $\text{det}(f) \in GL_1(R)$).

There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion $$ BGL_1(S^0) \to BGL_\infty(S^0) $$ where $BGL_1(S^0)$ is $BG$. Here $BGL_\infty(S^0)$ can be defined as the homotopy colimit over $k$ and $n$ of the topological monoid $G(\vee_k S^n)$ of homotopy automorphisms of a $k$-fold wedge of $n$-spheres.

Question 1: I've heard it mentioned that can be no retraction $A(*) \to BG$ to the map $BG \to A(\ast)$. Can someone please explain to me why there is no such splitting, and if possible, give a reference?

More generally,

Question 2: If $R$ is a structured ring spectrum, is there a reasonable set of conditions which guarantees that the map $BGL_1(R)\to K(R)$ admits a retraction?

(A related question is under what conditions does an $R$-module map $f: R^n \to R^n$ which is a weak homotopy equivalence admit a determinant $\text{det}(f) \in GL_1(R)$).