Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On regular genus and G-degree of PL 4-manifolds with boundary   In this article, we introduce two new PL-invariants: weighted regular genus
and weighted G-degree for manifolds with boundary. We first prove two
inequalities involving some PL-invariants which states that for any PL-manifold
$M$ with non spherical boundary components, the regular genus $\mathcal{G}(M)$
of $M$ is at least the weighted regular genus $\tilde{G}(M)$ of $M$ which is
again at least the generalized regular genus $\bar{G}(M)$ of $M$. Another
inequality states that the weighted G-degree $\tilde{D}_G (M)$ of $M$ is always
greater than or equal to the G-degree $D_G (M)$ of $M$. Let $M$ be any compact
connected PL $4$-manifold with $h$ number of non spherical boundary components.
Then we compute the following: $$\tilde{G} (M) \geq 2 \chi(M)+3m+2h-4+2 \hat{m}
\mbox{ and } \tilde{D}_G (M) \geq 12(2 \chi(M)+3m+2h-4+2 \hat{m}),$$ where $m$
and $\hat{m}$ are the rank of fundamental groups of $M$ and the corresponding
singular manifold $\hat{M}$ (obtained by coning off the boundary components of
$M$) respectively. As a
  consequence we prove that the regular genus $\mathcal{G}(M)$ satisfies the
following inequality:
  $$\mathcal{G} (M) \geq 2 \chi(M)+3m+2h-4+2 \hat{m},$$
  which improves the previous known lower bounds for the regular genus
$\mathcal{G}(M)$ of $M$. Then we define two classes of gems for PL $4$-manifold
$M$ with boundary: one consists of semi-simple and other consists of weak
semi-simple gems, and prove that the lower bounds for the weighted G-degree and
weighted regular genus are attained in these two classes respectively.

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The categories of this paper are: math-ph
3
The categories of this paper are: physics
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The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex