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How to construct closed, orientable, smooth, simply-connected $6$-manifolds such that $H^{*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p_{1}(M) = n a^{2}$? ($p_{1}(M) \in H^{4}(M,\mathbb{Z})$ denotes the first Pontryagin class)?

By Wall's classification of $6$-manifolds for each $n \in \mathbb{Z}$ exists a unique such manifold up to diffeomorphism. The case $n = 4$ is furnished by $\mathbb{C}\mathbb{P}^{3}$, are there at other familiar examples - perhaps the $n=0$ case? How about constructing these manifolds with surgery techniques?

How to construct closed, orientable, smooth, simply-connected $6$-manifolds such that $H^{*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p_{1}(M) = n a^{2}$? ($p_{1}(M) \in H^{4}(M,\mathbb{Z})$ denotes the first Pontryagin class).

By Wall's classification of $6$-manifolds for each $n \in \mathbb{Z}$ exists a unique such manifold up to diffeomorphism. The case $n = 4$ is furnished by $\mathbb{C}\mathbb{P}^{3}$, are there at other familiar examples - perhaps the $n=0$ case? How about constructing these manifolds with surgery techniques?

How to construct closed orientable, smooth, simply-connected $6$-manifolds such that $H^{*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p_{1}(M) = n a^{2}$? ($p_{1}(M) \in H^{4}(M,\mathbb{Z})$ denotes the first Pontrjagin class)?

By Wall's classification of $6$-manifolds for each $n \in \mathbb{Z}$ exists a unique such manifold up to diffeomorphism. The case $n = 4$ is furnished by $\mathbb{C}\mathbb{P}^{3}$, are there at other familiar examples - perhaps the $n=0$ case? How about constructing these manifolds with surgery techniques?

How to construct closed, orientable, smooth, simply-connected $6$-manifolds such that $H^{*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p_{1}(M) = n a^{2}$? ($p_{1}(M) \in H^{4}(M,\mathbb{Z})$ denotes the first Pontryagin class)?

By Wall's classification of $6$-manifolds for each $n \in \mathbb{Z}$ exists a unique such manifold up to diffeomorphism. The case $n = 4$ is furnished by $\mathbb{C}\mathbb{P}^{3}$, are there at other familiar examples - perhaps the $n=0$ case? How about constructing these manifolds with surgery techniques?