Homology Groups of Cubical Sets with Connections

Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of “connection” as a new kind of degeneracy. In this paper, for a cubical set with connections, we show that the connections generate an acyclic subcomplex of the chain complex of the cubical set. In particular, our results show that the homology groups of a cubical set with connections are independent of whether we normalize by the connections or we do not, that is, connections do not contribute to any nontrivial cycle in the homology groups of the cubical set.

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

The authors are grateful to Professor Ronald Brown for his valuable comments and suggestions on earlier versions of this paper. They also thank the referee for several useful remarks.

Authors and Affiliations

1. The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA, 94720, USA

   Hélène Barcelo

2. Haverford College, Haverford, PA, 19041, USA

   Curtis Greene

3. Department of Mathematics and Statistics, American University of Sharjah, PO Box 26666, Sharjah, United Arab Emirates

   Abdul Salam Jarrah

4. Fachbereich Mathematik und Informatik, Philipps-Universität, 35032, Marburg, Germany

   Volkmar Welker

Corresponding author

Correspondence to Abdul Salam Jarrah.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, USA.

Appendix: Homology of Cubical Sets and Homology of Their Geometric Realization

Recall that \(I^n\) is the geometric n-dimensional cube \([0,1]^n\). Let \((f_i^\alpha )^*: I^{n-1} \rightarrow I^n\) be the map sending \((x_1,\ldots ,x_{n-1}) \in I^{n-1}\) to \((x_1,\ldots , x_{i-1},y,x_{i},\ldots , x_{n-1})\) where \(y = 0\) if \(\alpha = -\) and \(y =1\) if \(\alpha = +\). Let further \((\varepsilon _i)^*: I^n \rightarrow I^{n-1}\) be the map sending \((x_1,\ldots , x_n)\) to \((x_1,\ldots , x_{i-1},x_{i+1},\ldots , x_n)\). The geometric realization |K| of a cubical set is the quotient space of the disjoint union \(\coprod I^n \times K_n\) by the equivalence relation \(\sim \), which is generated by the following elementary equivalences: For \((x_1,\ldots , x_n) \in I^n\) and \(\sigma \in K_{n-1}\) we set

and, for \((x_1,\ldots , x_{n-1}) \in I^{n-1}\) and \(\sigma \in K_{n}\), we set

Then |K| can be given the structure of a CW-complex whose (open) n-cells are the images \(e_\sigma ^{(n)}\) of the cells \(\mathring{I^n} \times \{\sigma \}\) in |K| for \(\sigma \in K^{nd}_n\). Here \(K^{nd}_n\) denotes the set of non-degenerate n-cubes in K, see [12, Remark 11.1.14]. Let S(K) be the cellular chain complex of |K|. By the definition of S(K) the cells \(e_\sigma ^{(n)}\) for \(\sigma \in K_n^{nd}\) form a basis of its n^th chain group \(S_n(K)\). It is well known (see [14, Corollary 3.9.11]) that identifying \(\sigma \in K_n^{nd}\) with \(e_\sigma ^{(n)}\) yields the following isomorphism of chain complexes.

Lemma 13

(Corollary 3.9.11 [14]) \({\mathcal {C}}(K) \cong S(K)\).

If the cubical set K is a cubical set with connections then there is an associated geometric realization \(|K|'\) which is the quotient of the disjoint union \(\coprod I^n \times K_n\) by the equivalence relation \(\sim '\), which is generated by (9), (10) and the relation

for \(\sigma \in K_{n-1}\) and \((x_1,\ldots , x_n) \in I^n\). Here \((\varGamma _i^\alpha )^*: I^{n} \rightarrow I^{n-1}\) is defined by

In particular, \(\sim '\) is coarser than \(\sim \) and hence \(|K|'\) can be seen as a quotient of |K| by the additional identifications implied by (11). Let \(K_n^{ndc}\) be the set of n-cubes in K that are neither degenerate nor connections.

In order to understand the relation between |K| and \(|K|'\) we need to understand the face structure of cubes in \(K_n^{ndc}\). For that we consider for any cube \(\sigma \in K_n\) the set of all of its faces \(\tau \); i.e. all cubes \(\tau \) such that \(\tau = f_{i_1}^{\alpha _1} ( \cdots (f_{i_r}^{\alpha _r}(\sigma ))\cdots )\) for a choice of \(i_1,\ldots , i_r\) and \(\alpha _1,\ldots , \alpha _r\). For \(\sigma \in K\) we denote by \(F_\sigma \) the set of its faces. We order the cubes from K by saying that \(\tau \) is smaller than \(\sigma \) if \(\tau \) is a face of \(\sigma \). With this notation we are in position to formulate the following structural result on the role of non-degenerate and non-connection cubes in the face structure.

Lemma 14

For any \(\tau \in K_{n}\) there is a unique face \(\rho \) of \(\tau \) that is maximal with the property that it is neither degenerate nor a connection. Moreover, if \(\tau = \varepsilon _i(\sigma )\) or \(\tau = \varGamma _i^\alpha (\sigma )\) then \(\rho \) is a subface of \(\sigma \) and \(\tau = g_k \cdots g_1 (\rho )\) for suitably chosen connection and degeneracy maps \(g_1,\ldots , g_k\) for some \(k \ge 0\).

Proof

We prove the assertion by induction on the dimension n.

If \(n = 0\) then \(\tau \) is non-degenerate and non-connection. Hence \(\tau \) itself is the maximal face we are looking for.

Let \(n > 0\). If \(\tau \) is neither degenerate nor a connection then again \(\tau \) itself is the unique maximal face.

Let \(\tau \) be degenerate, say \(\tau = \varepsilon _i(\sigma )\) for some \(i \in [n]\) and some \((n-1)\)-cube \(\sigma \). Then, by (iii) of Definition 1, \(f_j^\alpha (\tau ) = \sigma \) if \(i = j\), and \(f_j^\alpha (\tau ) = \varepsilon _{i-1}(f_j^\alpha (\sigma ))\) if \(j < i\) and \(= \varepsilon _{i}(f_{j-1}^\alpha (\sigma ))\) if \(j > i\). By induction, we know that there is an unique maximal non-degenerate and non-connection face \(\rho \) of \(\sigma \). We claim that \(\rho \) is the unique maximal non-degenerate and non-connection face of \(\tau \). By induction we know that each \(\varepsilon _r(f_s^\beta (\sigma ))\) has a unique maximal non-degenerate, non-connection face which is a subface of \(f_s^\beta (\sigma )\) and hence of \(\sigma \). In particular, they must be subfaces of \(\rho \). If follows by induction that \(\sigma = g_k\cdots g_1( \rho )\) for a sequence of degeneracy and connection maps \(g_1,\ldots , g_k\) and \(k \ge 0\). Then \(\tau = \varepsilon _i g_k \cdots g_1 (\rho )\).

Finally, consider the case that \(\tau \) is a connection. Say \(\tau = \varGamma _{i}^\alpha (\sigma )\) for some \(i \in [n]\) and some \((n-1)\)-cube \(\sigma \). Notice that, by (iii) of Definition 2, every \((n-1)\)-face of \(\tau \) other than \(\sigma \) is either \(\varGamma _j^\beta (f_t^\alpha (\sigma ))\) or \(\varepsilon _j(f_t^\alpha (\sigma ))\) for some \(j\in \), \(t\in [n]\), and \(\alpha \in \{+,-\}\). By induction \(\sigma \) and any \(\varGamma _j^\beta (f_t^\alpha (\sigma ))\) have an unique maximal non-degenerate, non-connection face. Again by induction the latter are subfaces of \(\sigma \). In particular, they must be subfaces of the unique maximal non-degenerate, non-connection face \(\rho \) of \(\sigma \). From the induction hypothesis it follows \(\sigma = g_k\cdots g_1 (\rho )\) for a sequence of degeneracy and connection maps \(g_1,\ldots , g_k\) and \(k \ge 0\). Then \(\tau = \varGamma _i^\alpha g_k \cdots g_1 (\rho )\). \(\square \)

Note that along the same lines one can show that for any cube there is a unique maximal non-degenerate face.

The relations among the degeneracy and connection maps allow the following strengthening of Lemma 14.

Lemma 15

For any \(\tau \in K_{n}\) there is a unique face \(\rho \) of \(\tau \) that is maximal with the property that it is neither degenerate nor a connection. Moreover, if \(\tau \) is non-degenerate then \(\tau = g_k \cdots g_1 (\rho )\) for suitably chosen connection maps \(g_1,\ldots , g_k\) and some \(k \ge 0\).

Proof

From Lemma 14 it follows that there is a unique maximal face \(\rho \) of \(\tau \) that is neither degenerate nor a connection. It also follows from that lemma that \(\tau = g_k\cdots g_1( \rho )\), for degeneracy and connection maps \(g_1,\ldots , g_k\). If all \(g_i\) are connection maps we are done. Assume there is an i such that \(g_i\) is a degeneracy map. We claim that then \(\tau \) is degenerate. We prove the claim by downward induction on the maximal i such that \(g_i\) is a degeneracy map. If \(i = k\) then \(\tau \) is degenerate, contradicting the assumptions. If \(i <k\) then by Definition 2(iii) there is a connection or degeneracy map \(g_i'\) and s degeneracy map \(g_{i+1}'\) such that

By induction this implies that \(\tau \) is degenerate. \(\square \)

Now we apply the results on the face structure in order to understand the attachment of cells in |K| and \(|K|'\). We assume without stating the proofs the following fact:

• Let \((x,\sigma ),(y,\sigma ) \in I^{\dim \sigma }\times \{\sigma \}\). Then \((x,\sigma )\), \((y,\sigma )\) are identified through the equivalence relation generated by 9,10 (resp. 9, 10 and 11) on \(\coprod _{\tau \in K} I^{\dim \tau } \times \{\tau \}\) if and only if they are identified by the equivalence relation generated by 9,10 (resp. 9, 10 and 11). on \(\coprod _{\tau \in F_\sigma } I^{\dim \tau } \times \{ \tau \}\).

This fact allows us to consider the identifications by the equivalence relations we consider as local identifications among points in the cells corresponding to the faces of a given cell.

Lemma 16

Let \(\tau \in K_n\) be such that \(\tau = g_k \cdots g_1 (\rho )\) for some cube \(\rho \) and connection maps \(g_1,\ldots , g_k\). Let \(\sim _\tau \) be the restriction of the equivalence relation generated by (9), (10), (11) to \(M_\tau = \coprod _{\sigma \in F_\tau } I^{\dim \sigma } \times \{\sigma \}\) and define \(\sim _\rho \) analogously. Then there is a retraction \(p_\tau : M_\tau /\sim _\tau \rightarrow M_\rho /\sim _\rho \).

Proof

We construct the retraction by induction on k. For \(k = 0\) the identity is the desired retraction.

Let \(k \ge 1\) and assume that for \(\tau ' = g_{k-1} \cdots g_1 \rho \) there is such a retraction \(p_{\tau '} : M_{\tau '}/\sim _{\tau '} \rightarrow M_\rho /\sim _\rho \). Then \(\tau = g_k \tau '\). The equivalence relation on \(I^{\dim \tau } \times \{\tau \}\) induced by the connection map \(g_k = \varGamma _i^\beta \) has equivalence classes being sets with fixed maximum or minimum of the i^th and \((i+1)\)^st coordinate depending on \(\beta \) being \(+\) or −. Each equivalence class has exactly two points that via the face maps \(f_i^\beta \) and \(f_{i+1}^\beta \) are identified with points in \(I^{\dim \tau '} \times \{ \tau '\}\), indeed both points are identified with the same point. The map that sends each equivalence class to the image of this point in \(M_{\tau '} /\sim _{\tau '}\) provides a retraction from \(M_\tau / \sim _\tau \) to \(M_{\tau '}/\sim _{\tau '}\). Composing this retraction with the retraction from \(p_{\tau '}\) provides the asserted retraction. This concludes the induction step. \(\square \)

We now introduce the concept of pushing cells for a general CW-complex which we will then match with the process of passing from |K| to \(|K|'\) in our case. Let X be a CW-complex where, for \(n \ge 0\), \(X_n = (e_\sigma ^{(n)})_{\sigma \in J_n}\) is the set of open n-cells in X for some indexing set \(J_n\). For each \(\sigma \in J_n\) let \(g_\sigma : \partial \overline{e^{(n)}} \rightarrow X^{(n-1)}\) be the attaching map. For some fixed \(N \ge 0\), let \(\bar{J}_N \subseteq J_N\) be a subset of the index set of the cells in dimension N such that, for each \(\sigma \in \bar{J}_N\),

• there is a \(\tau \in J_\ell \) for some \(\ell < N\) such that \(\mathfrak {I}g_\sigma \subseteq \overline{e^{(\ell )}_\tau }\), and

• for this \(\tau \) there is a retraction \(p_\sigma : \overline{e_{\sigma }^{(N)}} \rightarrow \overline{e_\tau ^{(\ell )}}\).

Now let \(X^{{\mathrm {push}}}\) be the CW-complex with \(X_n^{{\mathrm {push}}} = (\tilde{e}_\sigma ^{(n)})_{\sigma \in J_n'}\) the open n-cells in \(X^{{\mathrm {push}}}\) where \(J_n' = J_n\) for \(n \ne N\) and \(J_N' = J_N \setminus \bar{J}_N\) and attaching maps \(g'_\tau (x) = g_\tau (x)\) if \(g_\tau (x) \not \in \overline{e_\sigma ^{(N)}}\) for some \(\sigma \in \bar{J_N}\) and \(g'_\sigma (x) = p_\tau (g_\sigma (x))\) otherwise. In this situation we say that \(X^{{\mathrm {push}}}\) arises from X by pushing the cells \(e_\sigma ^{(N)}\) for \(\sigma \in \bar{J}_N\).

Next we show that |K| and \(|K|'\) are examples of CW-complexes that arise from each other by pushing cells.

Lemma 17

The geometric realization \(|K|'\) is a CW-complex that arises from the CW-complex of the geometric realization |K| by pushing the cells corresponding to connections successively by dimension in increasing order. In particular, \(|K|'\) can be given the structure of a CW-complex with n-cells indexed by the \(K_n^{ndc}\).

Proof

Since the first connection cells (that are not already degenerate) arise in dimension 2, we can assume the following situation. For some \(n \ge 2\) we have constructed a complex X such that

1. (a)

   X arises from |K| by pushing all cells that correspond to connections of dimensions \(< n\) where \(n \ge 2\).

2. (b)

   \(|K|/\sim _{n} \cong X\) where \(\sim _{n}\) is the equivalence relation which has singleton equivalence classes outside the closure of the cells of dimension \(<n\) and equals (11) when applied to the union of the closures of all other cells.

Now let \(\sigma \in K_n\) be a connection that is non-degenerate. Then by Lemma 14 there is a unique maximal face \(\tau \in K_{\ell }\) of \(\sigma \) which is non-degenerate and non-connection. Since all proper connection faces of \(\sigma \) have been pushed the attaching map \(g_\sigma \) of the N-cell \(I^N\) corresponding to \(\sigma \) has as its image the \(\ell \)-cell corresponding to \(\tau \). Furthermore, by Lemma 15 the conditions of Lemma 16 are satisfied and there is a retraction \(p_\sigma \) from then closure of the N-cell corresponding to \(\sigma \) to the closure of the \(\ell \)-cell corresponding to \(\sigma \). Moreover, by Lemma 16 the map \(\sigma \) identifies the exactly those elements which lie in the same equivalence class of \(\sim _n\).

Hence the conditions for a pushing to the cells corresponding to non-degenerate connections \(\sigma \) are satisfied. It follows that (a) and (b) are satisfied for \(\sim _n\). \(\square \)

Finally, we need to understand the impact of pushing cells on the cellular chain complex of a CW-complex.

Lemma 18

Let X be a CW-complex with cells \(X_n = (e_\sigma ^{(n)})_{\sigma \in J_n}\), \(n \ge 0\). Assume that there is a dimension N such that \(X^{{\mathrm {push}}}\) arises from X by pushing the cells \(e_\sigma ^{(N)}\) for \(\sigma \in \bar{J}_N \subseteq J_N\). Let

be the differential of the cellular chain complex associated to X. Then for \(\sigma \in J_n \setminus J_N\), \(\sigma ' \in J_{n-1} \setminus J_N\) the coefficient \(d^{{\mathrm {push}}}_{\sigma ,\sigma '}\) in the differential of the cellular chain complex of \(X^{{\mathrm {push}}}\) we have \(d^{{\mathrm {push}}}_{\sigma ,\sigma '} = d_{\sigma ,\sigma '}\).

Proof

The coefficient \(d_{\sigma ,\sigma '}\) is given as the degree of the composition

The composition depends on the attaching maps \(g_\sigma \) of the cells corresponding to \(\sigma \) only. Now consider the same sequence in \(X^{{\mathrm {push}}}\), which in particular implies \(\sigma ,\sigma ' \ne \tau \). Let \(g'_\sigma \) be the corresponding attaching maps. If \(g_\sigma (x) \not \in \overline{e_\tau ^{(N)}}\) for some \(\tau \in \bar{J}_N\) then \(g_\sigma (x) = g'_\sigma (x)\). If \(g_\sigma (x) \in \overline{e_\tau ^{(N)}}\) for some \(\tau \in \bar{J}_N\) then \(g'_\sigma (x) = p_\sigma (g_\sigma (x))\) for a retraction \(p_\sigma \). But in the latter case \(g_\sigma (x)\) and \(g'_\sigma (x)\) lie in the complement of any \((n-1)\) cell different from \(e_\tau ^{(N)}\). In that situation the composition is again determined by \(g_\sigma \). It follows that \(d_{\sigma ,\sigma '} = d^{{\mathrm {push}}}_{\sigma ,\sigma '}\). \(\square \)

By definition \({\mathcal {C}}_n(K)/{ {\text {Con}}}_n(K)\) has a basis indexed by \(K_n^{ndc}\). The differential of the complex \({\mathcal {C}}_n(K)/{ {\text {Con}}}_n(K)\) are arises from the differential in \({\mathcal {C}}(K)\) in the following way. Let \(\partial \alpha \) is the differential of \(\alpha \in K^{ndc}_n\) in \({\mathcal {C}}_n(K)\) then we set all coefficients of element from \(K_{n-1}^{nd} \setminus K_{n-1}^{ndc}\) to 0. Now the following theorem is an immediate consequence of Lemma 18 and Lemma 17.

Theorem 19

The cellular chain complex \(S'(K)\) of \(|K|'\) is isomorphic to the quotient complex \({\mathcal {C}}(K)/{ {\text {Con}}}(K)\). In particular,

Proof

The assertion follows immediately from Lemma 17 and Lemma 18. \(\square \)

The theorem together with Corollary 12 implies the following.

Corollary 20

Let K be a cubical set with connections. Then

This fact provides another motivation for the study of connections.

Reprints and permissions