No Access

A new invariant and decompositions of manifolds

Eiji Ogasa

Eiji Ogasa

Computer Science, Meijigakuin University, Yokohama, Kanagawa, 244-8539, Japan

E-mail Address: pqr100pqr100@yahoo.co.jp

E-mail Address: ogasa@mail1.meijigkauin.ac.jp

Search for more papers by this author

https://doi.org/10.1142/S0218216518500190Cited by:0 (Source: Crossref)

Abstract

We introduce a new topological invariant $ν (X) \in ℕ \cup {0}$ of compact manifolds-with-boundaries $X$ which is much connected with boundary-unions. A boundary-union is a kind of decomposition of compact manifolds-with-boundaries. See the body of the paper for the precise definition. Let $M$ and $N$ be $m$ -dimensional compact manifolds-with-boundaries. Let $M \cup_{\partial} N$ be a boundary-union of $M$ and $N$ . Then we have

0 ≦ ν (M ⋃ \partial N) ≦ max {ν (M), ν (N)} . <math display="block" altimg="eq-00009.gif"><mrow><mn>0</mn><mo class="MathClass-rel">≦</mo><mi>ν</mi><mfenced separators="" open="(" close=")"><mrow><mi>M</mi><munder class="msub"><mrow><mo mathsize="big">⋃</mo></mrow><mrow><mi>\partial</mi></mrow></munder><mi>N</mi></mrow></mfenced><mo class="MathClass-rel">≦</mo><mstyle><mtext class="textrm" mathvariant="normal">max</mtext></mstyle><mo class="MathClass-open" stretchy="false">{</mo><mi>ν</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>M</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><mi>ν</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>N</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">}</mo><mo class="MathClass-punc">.</mo></mrow></math>

We define

$ν (X)$ as follows: First, define an invariant of

$(n - 1)$ -closed manifolds. Take the maximum of the invariant of all connected-components of the boundary of each handle-body of an ordered-handle-decomposition with a fixed base

$A$ , where we impose the condition that the base

$A$ is a (not necessarily connected) closed manifold. Take the minimum of the maximum for all ordered-handle-decompositions with the base

$A$ . It is our another invariant

$ν (X, A)$ . Take the maximum of the minimum,

$ν (X,)$ , for all basis to satisfy the above condition. It is

$ν (X)$ . See the body of the paper for the precise definition.

Keywords:

AMSC: 57N10, 57N13, 57N15

References

1. W. Browder , Surgery on Simply-Connected Manifolds (Springer, 1972). Crossref, Google Scholar
2. M. B. Green, J. H. Schwarz and E. Witten , Superstring Theory, Vols. 1, 2 (Cambridge University Press, 1987). Google Scholar
3. R. C. Kirby , The Topology of 4-Manifolds (Springer, 1989). Crossref, Google Scholar
4. J. W. Milnor , Lectures on the h-Cobordism Theorem (Princeton University Press, 1965). Crossref, Google Scholar
5. E. Ogasa, A new invariant associated with decompositions of manifolds, preprints (2004, 2005), arXiv:hep-th/0401217, arXiv:math.GT/0512320. Google Scholar
6. E. Ogasa, Introduction to high dimensional knots, preprint (2013), arXiv:1304.6053 [math.GT]. Google Scholar
7. J. Polchinski , String Theory, Vols. 1, 2 (Cambridge University Press, 1998). Google Scholar
8. S. Smale , Generalized Poincaré conjecture in dimensions greater than four, Ann. of Math. 74 (1961) 391–406. Crossref, Web of Science, Google Scholar
9. C. Vafa et al., Mirror Symmetry (CMI/AMS publication, 2003). Google Scholar