# Injectivity of assembly in A-theory for $BC_2 = \mathbb R P^\infty$ in degree $4$

I am trying to understand the assembly map

$$\pi_i ((BC_2)_+ \wedge A( \ast )) \rightarrow A_i( BC_2 )$$

in low degrees for the space $$BC_2 = \mathbb R P^\infty$$ in Waldhausen $$A$$-theory. I know we have a splitting $$A( X ) \simeq \Sigma^\infty X_+ \vee \text{Wh}^{\text{DIFF}}(X)$$ and in the case of the point, $$\text{Wh}^{\text{DIFF}}(\ast)$$ is a 3-connective spectrum, with $$\text{Wh}_3^{\text{DIFF}}(\ast) = \mathbb Z/2$$ and $$\text{Wh}_4^{\text{DIFF}}(\ast) = 0$$. This means, the question of the injectivity of the assembly reduces to the assembly in $$\text{Wh}^{\text{DIFF}}$$, which makes it easy to show that the assembly is injective up till degree 3.

In degree 4 we're left with a term $$\pi_4 ((BC_2)_+ \wedge \text{Wh}^{\text{DIFF}}(\ast)) = H_1( C_2; \mathbb Z/2) = \mathbb Z/2$$. The image of this term seems to be of purely topological nature, as:

1. It, more or less by definition, doesn't come from the stable homotopy group $$\pi_4^{st} (BC_2)$$
2. It is killed under the morphism $$A( BC_2 ) \rightarrow A( \ast )$$ which is induced by the map that sends the generator $$t \in C_2$$ to $$-1$$.
3. It is killed under passage to $$K$$-theory of any discrete ring, i.e. the maps $$A( BC_2) \rightarrow K ( R [C_2])$$ can't detect it.

Which brings me to my question: Has $$A_4( BC_2 )$$ already been computed and what can we say about the image of the assembly?

Not an answer, but a possible approach: Using Dundas' cartesian square with corners $$A(*)$$, $$K(Z)$$, $$TC(*)$$ and $$TC(Z)$$ you can see that $$Wh^{Diff}_3(*) = Z/2$$ comes from $$TC_4(Z) = Z/8$$ (plus odd torsion) via the Mayer-Vietoris connecting homomorphism $$\partial$$. Hence $$H_1(BC_2; Wh^{Diff}_3(*)) = Z/2$$ also comes from $$H_1(BC_2; TC_4(Z)) = Z/2$$ via the homomorphism induced by $$\partial$$.
I think there is a commutative square with corners $$H_1(BC_2; TC_4(Z))$$, $$H_1(BC_2; A_3(*))$$, $$TC_5(Z[C_2])$$ and $$A_4(BC_2)$$. Hence you might try to calculate the assembly map $$H_1(BC_2; TC_4(Z)) \to TC_5(Z[C_2])$$. If its image in $$TC_5(Z[C_2])$$ is nonzero, and not in the combined image from $$K_5(Z[C_2])$$ and $$TC_5(BC_2)$$, then the assembly map to $$A_4(BC_2)$$ will be injective.
I think calculating the 2-torsion in $$TC_5(Z[C_2])$$ (or $$K_5(Z_2[C_2])$$) is the harder part here. Understanding $$TC_5(BC_2)$$ should be easy from the Boekstedt-Hsiang-Madsen formula. Identifying the image from $$K_5(Z[C_2])$$ might also be hard.