In this question I have asked about boundary of regular neighborhood of $\mathbb RP^2$ in $\mathbb R^4$. I am interested in more general way of producing 3-manifolds in $\mathbb R^4$ namely the boundaries of regular neighborhood of 2-dimensional CW-complex.

In below article it is proven that every 2-complex can be embedded in 4-dimensional Euclidean space up to homotopy type.

Dranišnikov, A. N.; Repovš, Dušan, Embedding up to homotopy type in Euclidean space, Bull. Aust. Math. Soc. 47, No. 1, 145-148 (1993). ZBL0796.57011.

Let $K$ be such a complex which contain bouquet of $n$ circles $a_1,...,a_n$ as 1-skeleton and $k$ two-dimensional cells $e_1,...,e_k$ attached along words $w_1,...,w_k$. Le $M$ denote the boundary of regular neighborhood of $K$. In order to calculate homology group $H_1(M)$, I need to add circles $S^1(e_i)$ for $i=1,...,k$. Such circle surround interior of cell $e_i$ which is well defined in 4-dimensional space. Now the question is what additional relations we should add for the group.

Consider example of 2-complex with two words and two 2-cells attached along words $a^2b^2, a^{-2}b^3$. Then homology is calculated as

$H_1=\frac{Z+Z+Z}{\begin{bmatrix}2 & -2 & 0 \\2 &3 &0\\0& 0& 2 \\ \end{bmatrix}}=Z_2+Z_2+Z_5$

The third generator come from 2-cell for first word. Second word generator is trivial, because we have relations $d^{-2}, d^3$. My guess for relation come from having the circles $S^1(e)$ approaching cell boundary for each generator circle from 1-skeleton. There must be some mistake in my guess, because resulting homology should have torsion of shape $G+G$. It is conclusion from below paper:

Hantzsche, W., Einlagerung von Mannigfaltigkeiten in euklidische Räume., Math. Z. 43, 38-58 (1937). ZBL63.0556.02.

My question is how to obtain homology group of manifold $M$ obtained in described way from 2-complex $K$ embedded in $\mathbb R^4$ ?

I have asked the same question on math.stackexchage but without any answer or comment, so I try here.

Second related question is following. Say that I start building my 2-complex neighborhood step by step starting from 1-skeleton and attaching 2-cells. The neighborhood of bouquet of circles is connected sum of $n$ copies of $S^1\times S^2$. Next we fix $k$ circles on the boundary corresponding to words $w_1..w_k$. Finally we attach handles along these circles. Is result depending on the way we select $k$ circles ?

  • $\begingroup$ On the second question: yes, the result depends on the way you select the circles. For instance, every Mazur manifolds is the neighbourhood of a (contractible) 2-complex with one cell of each dimension $\le 2$, but in general they do not have boundary $S^3$. $\endgroup$ – Marco Golla Sep 17 '18 at 20:48

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