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| Main Authors: | , |
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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2103.13602 |
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| _version_ | 1866911316901363712 |
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| author | Basak, Biplab Binjola, Manisha |
| author_facet | Basak, Biplab Binjola, Manisha |
| contents | In this article, we study a class of closed connected orientable PL $4$-manifolds admitting a semi-simple crystallization and which have an infinite cyclic fundamental group. We show that the manifold in the class admits a handle decomposition in which the number of $2$-handles depends upon its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$. More precisely, our main result is the following. For a closed connected orientable PL $4$-manifold having a semi-simple crystallization with the fundamental group as $\mathbb{Z}$, we have constructed a handle decomposition for $M$ as one of the following types:
$(1)$ one $0$-handle, two $1$-handles, $1+β_2(M)$ $2$-handles, one $3$-handle and one $4$-handle,
$(2)$ one $0$-handle, one $1$-handle, $β_2(M)$ $2$-handles, one $3$-handle and one $4$-handle,
where $β_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_13602 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Handle decompositions for a class of closed orientable PL 4-manifolds Basak, Biplab Binjola, Manisha Geometric Topology Combinatorics Primary 57Q15, Secondary 05C15, 05E45, 57Q05, 57M15, 57M50 In this article, we study a class of closed connected orientable PL $4$-manifolds admitting a semi-simple crystallization and which have an infinite cyclic fundamental group. We show that the manifold in the class admits a handle decomposition in which the number of $2$-handles depends upon its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$. More precisely, our main result is the following. For a closed connected orientable PL $4$-manifold having a semi-simple crystallization with the fundamental group as $\mathbb{Z}$, we have constructed a handle decomposition for $M$ as one of the following types: $(1)$ one $0$-handle, two $1$-handles, $1+β_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, $(2)$ one $0$-handle, one $1$-handle, $β_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, where $β_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients. |
| title | Handle decompositions for a class of closed orientable PL 4-manifolds |
| topic | Geometric Topology Combinatorics Primary 57Q15, Secondary 05C15, 05E45, 57Q05, 57M15, 57M50 |
| url | https://arxiv.org/abs/2103.13602 |