In constructing singular homology for a topological space, the boundary operator for the singular chain complex is given as an alternating sum of face maps. The degeneracy maps seem to be discarded in converting a simplicial object into a differential-graded object. Out of curiosity, I took the alternating sum of (the linear maps generated by) the degeneracy maps and was pleasantly surprised to find that the examples I considered squared to zero like a coboundary.


Does the alternating sum of degeneracy maps yield the coboundary for a cochain complex on singular chains? If so, is there a name for the cohomology of this complex? Googling a few obvious phrases has given me nothing to work with.



A cochain complex using degeneracy maps
Recall the following categories:


* $\Delta$, the "simplicial bookkeeping category" of nonempty finite linearly ordered sets
* $\Delta\_+$, the category of possibly empty finite linearly ordered sets obtained by adjoining an initial object to $\Delta$
* $\Delta\_{-\infty}$, the category of nonempty finite linearly ordered sets and minimum-preserving maps
* $\Delta\_{\pm\infty}$, the category of nonempty finite linearly ordered sets and minimum-and-maximum preserving maps


Evidently, we have faithful functors $\Delta\_{\pm\infty}\to\Delta\_{-\infty}\to\Delta\to\Delta\_+\to \Delta\_{-\infty}$, where the last map adds a new minimum to the partially ordered set.


Functors from (the opposites of) these categories to abelian groups have the following names:


1. A **simplicial abelian group** $A\_\bullet:\Delta^{op}\to \operatorname{Ab}$ determines a normalized chain complex $N\_\*(A\_\bullet)$ (in fact, this is an equivalence of categories)
2. An **augmented simplicial abelian group** $A\_\bullet:\Delta^{op}\_+\to\operatorname{Ab}$ determines a map $H\_0(N\_\*(A\_\bullet))\to A(\emptyset)$ (in fact, this is an equivalence of categories)
3. A **split simplicial abelian group** $A\_\bullet:\Delta^{op}\_{-\infty}\to\operatorname{Ab}$ determines a strong chain deformation retraction $(A(\emptyset)\to N\_\*(A\_\bullet),h:N\_\*(A\_\bullet)\to N\_{\*+1}(A\_\bullet))$ of the associated augmented simplicial object (this should also be an equivalence of categories).
4. Finally, the category $\Delta\_{\pm\infty}$ is equivalent to $\Delta\_+$ by sending a function $f:[n]\to [m]$ to its "effect on gaps", i.e. the map $\widetilde f:[m-1]\to [n-1], \widetilde f(j) = \max \{i\mid f(i) \le j\}$. Thus a functor $A\_\bullet:\Delta\_{\pm\infty}^{op}\to\operatorname{Ab}$ is the same data as a **cosimplicial abelian group**, which in turn form a category equivalent to augmented cochain complexes.


Your construction takes a simplicial ableian group $A\_\bullet$ and essentially produces the mapping cone of the augmentation $A\_0 = B(\emptyset)\to N^\*(B^\bullet)$ associated to the augmented cosimplicial group $B^\bullet$. It is straightforward to show that the "effect on gaps" construction extends to a commutative diagram
$$
\require{AMScd}
\begin{CD}
\Delta\_{\pm\infty} @>{\cong}>> \Delta\_+^{op}\\
@VVV @VVV\\
\Delta\_{-\infty}@>{\cong}>>\Delta\_{+\infty}^{op}\cong \Delta\_{-\infty}^{op}
\end{CD}
$$
where $\Delta\_{+\infty}$ is the subcategory of maximum-preserving monotone functions (the equivalence to $\Delta\_{-\infty}$ is given by reversing the partial order.) Since $A\_\bullet$ extends over $\Delta\_{-\infty}$, so does $B^\bullet$, which shows that this mapping cone is equipped with a canonical chain nullhomotopy. (This is essentially the "extra degeneracy" argument of Tom Goodwillie's comment.)


In fact, this works in any abelian category, even if chain complexes with vanishing homology don't necessarily possess a chain homotopy retraction. This should show that the Whitehead torsion mentioned by Connor Malin vanishes.


(In fact)${}^2$, this works in any $\infty$-category: augmented simplicial objects are (homotopy) colimit cones under the corresponding simplicial object, and split simplicial objects are **absolute (homotopy) colimit cones** in the sense that the underlying augmented simplicial object gets sent to a (homotopy) colimit cone by any functor. This is shown in Lemma 6.3.1.16 of Lurie's Higher Topos Theory.


