In Barden's 1965 paper: Simply-connected five manifolds, Barden gave a complete list of diffeomorphism classes of simply-connected 5-manifolds: $$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$ where $-1\le j\le\infty$, $1<k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1}=\infty$ and $\#$ denotes the connected sum of oriented manifolds. The manifold $X_{j',k_1',\dots,k_n'}$ is diffeomorphic to $X_{j,k_1,\dots,k_n}$ if and only if $(j',k_1',\dots,k_n')=(j,k_1,\dots,k_n)$.

Following Barden's notation, let $A \cup B$ denote the disjoint union of the two manifolds $A^n$, $B^n$. If $A$ and $B$ have non-empty boundaries $\partial A$, $\partial B$, then $A + B$ is formed from $A \cup B$ by embedding $(n- 1)$-discs in $\partial A$ and $\partial B$, identifying them under an orientation reversing diffeomorphism (the orientations of the embedded discs being induced from those of $\partial A$ and $\partial B$), and smoothing the corners. More generally we form $A + fB$, where $f$ is an orientation reversing diffeomorphism of any $(n - 1)$-dimensional submanifold of $\partial B$ with one of $\partial A$. Note that $\partial(A+B)=\partial A\#\partial B$.

Let $M$ be an oriented 5-manifold and $f$ an orientation preserving diffeomorphism of $\partial M$ onto itself. Then, if $M^*$ is a second copy of $M$ but with the opposite orientation, $f$ may be regarded as an orientation reversing diffeomorphism of $\partial M^*$ onto $\partial M$ and we may form the oriented closed manifold $M+ f M^*$. We shall require $f$ to realize given automorphisms $f_*$ of $H_2(\partial M)$.

Now let $A_1$ and $A_2$ be two copies of $D^3\times S^2$, $M_k$ is defined as $$M_k=(A_1+A_2)+f_k(A_1+A_2)^*$$ where $f_k$ realizes the automorphism $$(f_k)_*(a_1,b_1,a_2,b_2)=(a_1,b_1+ka_2,a_2,b_2-ka_1).$$ My question: What is the cell structure of $M_k$? (I have found the cell structure of $X_j$ in Barden's paper.)


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