# Cell structures of simply-connected 5-manifolds (classified by Barden's 1965 paper)

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$$ divides $$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.)