group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
By the discussion at cohomology, plain cohomology is the study of
(∞,1)-categorical hom-spaces in (∞,1)-toposes (for “nonabelian cohomology”)
or their stabilizations to stable (∞,1)-categories of spectrum objects (for “generalized (Eilenberg-Steenrod) cohomology”)
or generally in symmetric monoidal (∞,1)-categories
and maybe fully generally in any (∞,1)-category $\mathcal{C}$ whatsoever.
So for $A \in \mathcal{C}$ any object the cohomology of any other object $X$ with coefficients in $A$ is the mapping space $\mathcal{C}(X,A)$. Notice that this is equivalently the homotopy type of sections $\mathcal{C}_{/X}(X, X \times A)$ of the trivial $A$-fiber ∞-bundle over $X$. The idea of twisted cohomology then is to consider general $A$-fiber ∞-bundles $\chi$ over $X$ and take the $\chi$-twisted cohomology of $X$ to the type of sections of this.
cohomology | twisted cohomology |
---|---|
homotopy types of mapping spaces | homotopy types of spaces of sections |
Given an $\infty$-topos $\mathbf{H}$, then also its arrow $\infty$-category $\mathbf{H}^I$ is an $\infty$-topos, over $\infty Grpd^I$ and it also sits over $\mathbf{H}$ by the codomain fibration, constituting an “extension” of $\mathbf{H}$ by itself:
The intrinsic cohomology of $\mathbf{H}^I$ under this fibration is nonabelian twisted cohomology as discussed in some detail in Principal ∞-bundles -- theory, presentations and applications.
Notice that “stable cohomology”, which is traditionally called generalized (Eilenberg-Steenrod) cohomology may be thought of as the lowest order Goodwillie approximation to nonabelian cohomology: where a cocycle in nonabelian cohomology is a map to any homotopy type, a cocycle in generalized (Eilenberg-Steenrod) cohomology is a map into a stable homotopy type.
In this sense the tangent (infinity,1)-topos $T \mathbf{H}$ is the lowest order linear approximation to the codomain fibration
Higher-order approximations should involve a notion of higher-order forms of the tangent (∞,1)-topos, in parallel with the relationship between the jet bundles and tangent bundle of a manifold. It is clear that whatever we may say in detail about the $k$th-jet (∞,1)-topos $J^k \mathbf{H}$, its intrinsic cohomology is a version of twisted cohomology which is in between nonabelian cohomology and stable i.e. generalized (Eilenberg-Steenrod) cohomology.
It seems that a layered analysis of nonabelian cohomology this way in higher homotopy theory should eventually be rather important, even if it hasn’t received any attention at all yet. It seems plausible that a generalization of Chern-Weil theory which approximates classes of principal infinity-bundles not just by universal characteristic classes in ordinary cohomology and hence in stable cohomology, but that one wants to consider the whole Goodwillie Taylor tower of approximations to it.
We discuss concrete realizations of the above general idea in some cases of interest:
Let $\mathcal{C} = \mathbf{H}$ be an (∞,1)-topos. Let $A \in \mathbf{H}$ be any object, to be called the coefficient object.
Write $\mathbf{Aut}(A) \in Grp(\mathbf{H})$ for the automorphism ∞-group of $A$ and $\mathbf{B}\mathbf{Aut}(A) \in \mathbf{H}$ for its delooping. There is a canonical ∞-action of $\mathbf{Aut}(A)$ on $A$ exhibited by the corresponding universal associated ∞-bundle
Let $X \in \mathbf{H}$ be any object.
A twist for $A$-cohomology on $X$ is a morphism $\chi \colon X \to \mathbf{B}\mathbf{Aut}(A)$ in $\mathbf{H}$. The corresponding associated $A$-fiber ∞-bundle over $X$ which is the homotopy pullback
we call the local coefficient ∞-bundle for twisted $A$-cohomology classified by $\chi$.
The cocycle ∞-groupoid of $\chi$-twisted $A$-cohomology is
The $\chi$-twisted cohomology set of $X$ is
Special cases of this definition are implicit in traditional literature. The above statement appears in this form in (Nikolaus-Schreiber-Stevenson 12).
The $\chi$-twisted cohomology is equivalently the ordinary cohomology of $\chi$ with coefficients in $\rho_A$ in the slice (∞,1)-topos of $\mathbf{H}$ over $\mathbf{B}\mathbf{Aut}(A)$:
Let now $\mathcal{C} = Stab(\mathbf{H})$ be an stable (∞,1)-category of spectrum objects in an ambient (∞,1)-topos $\mathbf{H}$. Let $E \in CRing_\infty(\mathbf{H})$ be a corresponding E-∞ ring object. Write
for the ∞-group of units of $E$.
Now a twist $\chi \;\colon\; X \to \mathbf{B}GL_1(E)$ classifies an (∞,1)-module bundle of $E$-lines. The $\chi$-twisted $E$-cohomology is again the (stable) homotopy type of sections of this.
For the case of twisted K-theory (see the references there) this description goes back to Jonathan Rosenberg. The above general abstract description is developed in (Ando-Blumberg-Gepner 10).
For more details see
at (∞,1)-module bundle the section Sections and twisted cohomology
at twisted bivariant cohomology the section Axiomatization in homotopy theory .
There are canonical maps
where $Pic(E Mod)$ denotes the Picard ∞-groupoid. This suggest to speak not just of twists of the form $\chi \colon X \to \mathbf{B}GL_1(E) \simeq E Line \hookrightarrow E Mod$ but more generally of twists of the form $\chi \colon Pic(E Mod) \hookrightarrow E Mod$. While these in general no longer define $E$-fiber ∞-bundles (so that sections of them are strictly speaking in general no longer locally $E$-cohomology cocycles), this more general notion has the advantage that it makes sense also in symmetric monoidal (∞,1)-categories different from those of the form $Stab(\mathbf{H})$.
This we turn to below.
If in the above situation we write $[X, E Mod]$ for the symmetric monoidal (∞,1)-category of $E$-(∞,1)-module bundles on $X$, then given an object $\chi \in [X,E Mod]$ its homotopy type of sections, hence the $\chi$-twisted cohomology of $X$ is equivalently
where $\mathbb{I}_X$ is the tensor unit object, the trivial $E$-(∞,1)-module bundle over $X$.
In view of this and remark 2 one considers the following.
Let $(\mathcal{C}, \otimes)$ be a symmetric monoidal (∞,1)-category.
An object $\chi \in Pic(\mathcal{C})$ of the Picard ∞-groupoid of $\mathcal{C}$ we call a twist for cohomology in $\mathcal{C}$. For $X, A \in \mathcal{C}$ any two objects, we say that the $\chi-twisted$ cohomology of $X$ with coefficients in $A$ is
old material, to be harmonized…
Let ${*} \to B$ be a pointed object. Then
we say that the cocycle
$(X \to * \to B) \in \mathbf{H}(X,B)$
is the trivial $B$-cocycle on $X$.
the morphism $f:\hat{B}\to B$ induces a fibration sequence
$A \to \hat B \stackrel{f}{\to} B$
in $\mathbf{H}$.
The $([*],f)$-twisted cohomology with trivial twisting cocycle is equivalent to the ordinary cohomology with coefficients in the homotopy fiber $A$ of $f$:
By definition, the homotopy fiber of $A$ is the homotopy pullback
in $\mathbf{H}$. Since the $\infty$-groupoid valued hom in an (∞,1)-category is exact with respect ot homotopy limits (by definition of homotopy limits), it follows that for every object $X$, there is fibration sequence of cocycle ∞-groupoids
By definition of twisted cohomology, this identifies
For this reason, when $B$ is pointed, it is customary to call the set of equivalence classes $\pi_0\mathbf{H}_{[c]}(X;f)$ the $c$-twisted $A$-cohomology of $X$, and to denote it by the symbol
The cohomology fibration sequence $\mathbf{H}(X,A) \to \mathbf{H}(X,\hat B) {\to} \mathbf{H}(X,B)$ can be seen as an obstruction problem in cohomology:
But it also says:
For $V$ a vector space and $X$ a manifold, both regarded a 0-truncated objects in the $(\infty,1)$-topos on the site CartSp (that of Lie infinity-groupoids), a cocycle $X \to V$ is simply smooth $V$-valued function on $X$.
Now let $G$ be a Lie group with smooth delooping groupoid $\mathbf{B}G$ and let $\rho : \mathbf{B}G \to Vect$ be a representation of $G$ on $V$, i.e. $\rho(\bullet) = V$. Then the corresponding action groupoid $V//G$ sits in the fibration sequence
Hence we can ask for the $p$-twisted cohomology of $X$ with values in $V$. Now, a cocycle $g : X \to \mathbf{B}G$ is one classifying a $G$-principal bundle on $X$. By looking at this in Cech cohomology it is immediate to convince onself that cocycles $X \to V//G$ such that the composite $X \to V//G \stackrel{p}{\to} \mathbf{B}G$ is equivalent to the given $g$ are precisely the sections of the $\rho$-associated vector bundle:
on a patch $U_i$ of a good cover over wich $P$ has been trivialized, the cocycle $X \to V//G$ is simply a $V$-valued function $\sigma_i : U_i \to V$. Then on double overlaps it is a smooth natural transformation $\sigma_i|_{U_{i j}} \to \sigma_j|_{U_i j}$ whose components in $G$ are required to be those of the given cocycle $g$. That means exactly that the functions $(\sigma_i)$ are glued on double overlaps precisely as the local trivializations of a global section $\sigma : X \to P \times_G V$ would.
Hence we find the $p$-twisted cohomology is
In this sense a section is a twisted function.
Notice that $V//G \stackrel{p}{\to} \mathbf{B}G$ is not itself a homotopy fiber, but is a lax fiber, in that we have a lax pullback (really a comma object )
where in the bottom right corner we have Vect (regarded as a stack on $CartSp$ in the evident way) and where the right vertical morphism sends the point to the ground field vector space $k$ (or rather sends $U \in CartSp$ to the trivial vector bundle $U \times k$ ).
We may paste to this the homotopy pullback along the cocycle $g : X \to \mathbf{B}G$ to obtain
This makes is manifest that a section $\sigma : X \to P \times_G V$ is also the same as a natural transformation from $const_k : X \to Vect$ to $X \stackrel{g}{\to} \mathbf{B}G \to Vect$.
Notice moreover that in the special case that $G = U(1)$ and for ground field $k = \mathbb{C}$ we may think of $\mathbf{B}U(1)$ as the category $\mathbb{C} Line \hookrightarrow \mathbb{C} Mod = Vect$ and think of the twisting cocycle $g$ as
Regarded this way, the above discussion has a generalization to the case where the monoid $\mathbb{C}$ is replaced with any ring spectrum $R$ and we consider
Twisted cohomology in terms of such morphisms $\tau$ is effectively considered in
and in unpublished work of Ulrich Bunke et al. For more on this see the discussion at (∞,1)-vector bundle.
More generally one can hence twist with maps
into the Picard ∞-group of $R Mod$.
See also at ∞-group of units – augmented definition.
In the context of generalized (Eilenberg–Steenrod) cohomology a coefficient object for cohomology is a spectrum $A$: the $A$-cohomology of a topological space $X$ with coefficients in $A$ is the set of homotopy classes of maps $X \to A$. For instance, as a model of the degree-$0$ space in the K-theory spectrum one can take $A = Fred(H)$, the space of Fredholm operators on a separable Hilbert space $H$. There is a canonical action on this space of the projective unitary group $G = P U(H)$ of $H$. Since $P U(H)$ has the homotopy type of an Eilenberg–Mac Lane space $K(\mathbb{Z},2)$, a $P U(H)$-principal bundle $P \to X$ defines a class $c \in H^3(X,\mathbb{Z})$ in ordinary integral cohomology (this may also be thought of as the class of a twisting bundle gerbe). The twisted K-theory (in degree $0$) of $X$ with that class as its twist is the set of homotopy classes of sections $X \to P \times_{P U(H)} Fred(H)$ of the associated bundle.
The above example generalizes straightforwardly to the case that
$A$ is a connective spectrum, i.e. topological space that is an infinite loop space. (We need to assume a connective spectrum given by an infinite loop space so that $A$ can be regarded as living in the category of topologicall spaces along with the other objects, such as classifying spaces $\mathbf{B}G$ of nonabelian groups);
with a (topological) group $G$ acting on $A$ by automorphisms and
a $G$-principal bundle $P \to X.$
In this case there is an established definition of generalized (Eilenberg–Steenrod) cohomology with coefficients $A$ twisted by a $G$-principal bundle as follows.
From the $G$-principal bundle $P \to X$ we obtain the associated $A$-bundle $P \times_G A \to X$. Then a twisted $A$-cocycle on $X$ is defined to be a section $X \to P \times_G A$ of this associated bundle. The collection of homotopy classes of these sections is the twisted $A$-cohomology of $X$ with the twist specified by the class of $P$.
This is the definition of twisted cohomology as it appears for instance essentially as definition 22.1.1 of the May–Sigursson reference below (when comparing with their definition take their $G$ to be the trivial group and identify their $\Gamma$ and $\Pi$ with our $G$).
It is clearly a particular case of the general definition of twisted cohomology given above:
the $(\infty,1)$-topos $\mathbf{H}$ is the $(\infty,1)$-category of Top of topological spaces
the object $B$ is the delooping $\mathbf{B}G$ of the group $G$.
the object $\hat{B}$ is the homotopy quotient $A//G\simeq \mathbf{E}G\times_G A$.
the twisting cocycle $c$ is the element in $\mathbf{Top}(X,\mathbf{B}G)$ defining the principal $G$-bundle $P\to X$.
Indeed, $B$ is pointed, we have a fibration sequence
and the homotopy pullback
is the $A$-bundle $P\times_G A\to X$.
The obstruction problem described by this example reads as folllows:
Read the other way round it says:
Since the associated bundle $P \times_G A$ is in general no longer itself a spectrum, twisted cohomology is not an example of generalized Eilenberg–Steenrod cohomology.
To stay within the spectrum point of view, May–Sigurdsson suggested that twisted cohomology should instead be formalized in terms of parameterized homotopy theory, where one thinks of $P \times_G A$ as a parameterized family of spectra.
Some somewhat trivial examples of this appear in various context. For instance group cohomology on a group with coefficients in a nontrivial module can be regarded as an example of twisted cohomology. See there for more details.
Compare this to the example below of cohomology “with local coefficients”. It is the same principle in both cases.
To get a feeling for how the definition does, it is instructive to see how for the fibration sequence coming from an ordinary central extension $K \to \hat G \to G$ of ordinary groups as
classified by a group 2-cocycle $\omega$, $c$-twisted $\hat G$-cohomology produces precisely the familiar notion of twisted bundles, with the twist being the lifting gerbe that obstructs the lift of a $G$-bundle to a $\hat G$-bundle.
This is also the first example in the list in the last section of
and contains examples that are of interest in the wider context of string theory.
See also Twisted Differential String- and Fivebrane-Structures.
What is called cohomology with local coefficients is twisted cohomology with the twist given by the class represented by the universal cover space of the base space, which is to say: by the action of the fundamental group of the base space.
In the classical case of ordinary cohomology, C. A. Robinson in 1972 constructed a twisted $K(\pi,n)$, denoted $\tilde K(\pi,n)$, so that, for nice spaces, the cohomology with local coefficients $\tilde H^n(X,\pi)$ with respect to a homomorphism $\varepsilon:\pi_1(X)\to Aut(\pi)$ is given by homotopy classes of maps $X\to \tilde K(\pi,n)$ compatible with $\varepsilon.$
More generally, for any space $X$, let $A$ be a coefficient object that is equipped with an action of the first fundamental group $\pi_1(X)$ of $X$. (Such an action is also called an $A$-valued local system on $X$).
Then there is the fibration sequence
of this action.
Notice that there is a canonical map $c : X \to \mathbf{B} \pi_1(X)$, the one that classifies the universal cover of $X$.
Then $A$-cohomology with local coefficients on $X$ is nothing but the $c$-twisted $A$-cohomology of $X$ in the above sense.
By effective, we mean involving as much as possible only calculations within finite dimensional linear algebra. For definiteness, we work in the smooth context and require the locally constant sheaf $\mathcal{A}$ to have stalks of finite dimensional vector spaces over a field $k$ ($\mathbb{R}$ or $\mathbb{C}$). Let $X$ be a connected $n$-dimensional manifold. The sheaf $\mathcal{A}$ can then be seen as the sheaf of germs of locally constant sections of a vector bundle $A\to X$ endowed with a flat connection $\nabla$. The fibers of $A$ are isomorphic with the stalks of $\mathcal{A}$, all of which are isomorphic to some finite dimensional vector space $\bar{A}$. Let $\tilde{X} \to X$ denote the universal cover of $X$ and $\pi = \pi_1(X)$ its fundamental group. It is well known that $\pi$ acts by deck transformation diffeomorphisms on $\tilde{X}$ and also induces a holonomy representation $\rho\colon \pi \to \mathbf{Aut}(\bar{A})$ on $\bar{A}$.
Consider the vector bundles $\Lambda^p X \otimes_X A$, where $\Lambda^p X$ is the bundle of differential $p$-forms, with $\Omega^p_A(-)$ denoting the sheaf of its sections (differential forms twisted by $A$). Let $d_\nabla \colon \Omega^p_A \to \Omega^{p+1}_A$ denote the correspondingly twisted de Rham differential, defined by the property that
where $d$ is the ordinary de Rham differential, $\nabla a$ is seen as a section of $\Lambda^1 X \otimes_X A$, and with the wedge operation acting in the obvious way. The complex of sheaves $(\Omega^\bullet_A(-),d_\nabla)$ is then a soft sheaf? resolution of the sheaf $\mathcal{A}$ of locally constant sections of $A$. Its cohomology $H^p(X;A,\nabla) = H^p(\Omega^\bullet_A(X),d_\nabla)$ is then isomorphic to the sheaf cohomology of $X$ with coefficients in the locally constant sheaf $\mathcal{A}$, $H^p(X,\mathcal{A}) \simeq H^p(X;A,\nabla)$.
Now, the bundle $A\to X$ and the connection $\nabla$ both pull back to the universal covering space $\tilde{X}$, that is to $\tilde{A} \to \tilde{X}$ and $\tilde{\nabla}$. Since now $\tilde{X}$ is simply connected, we can globally trivialize this bundle as $\tilde{A} \simeq \tilde{X} \times \bar{A}$ and $\tilde{\nabla}$ to the trivial connection thereon. Similarly, the structure of the sheaf of twisted differential forms can be simplified to $\Omega^p_{\tilde{A}}(-) \simeq \Omega^p(-) \otimes \bar{A}$, with the action of the twisted de Rham differential given by $d_{\tilde{\nabla}} (\omega \otimes \bar{a}) = d\omega \otimes \bar{a}$. This observation allows us to conclude that, on the universal covering space, we have the isomorphism $H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}$, where $H^p(\tilde{X})$ denotes the ordinary de Rham cohomology.
The pull back along a deck transformation diffeomorphism, induces a linear action of $\pi$ on forms $\Omega^p(\tilde{X})$. Combined with the holonomy representation of $\pi$ on $\bar{A}$, this defines a representation of $\pi$ on $\Omega^p_{\tilde{A}}(X)$. Let $\Omega^p_{\tilde{A}}(\tilde{X})^\pi \subseteq \Omega^p_{\tilde{A}}$ denote the subspace of twisted forms that is invariant under the action of $\pi$. It is not hard to notice the isomorphism of complexes $(\Omega^\bullet_\tilde{A}(\tilde{X})^\pi, d_{\tilde{\nabla}}) \simeq (\Omega^\bullet_A(X), d_\nabla)$ and hence of their cohomologies, $H^p(X; A,\nabla) \simeq H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^\pi, d_{\tilde{\nabla}})$. Furthermore, the action of $\pi$ on $\Omega^p_{\tilde{A}}(\tilde{X})$ commutes with the differential $d_{\tilde{\nabla}}$ and hence induces an action on $H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}$, where $\pi$ also acts in the obvious and compatible way on each tensor factor. Let $(H^p(\tilde{X})\otimes \bar{A})^\pi$ denote the corresponding $\pi$-invariant subspace.
Suppose that there exists a decomposition $\Omega^p_{\tilde{A}}(\tilde{X}) \simeq \Omega^p_{\tilde{A}}(\tilde{X})^\pi \oplus \Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}}$ as representations of $\pi$, with $\Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}}$ having no $\pi$-invariant subspace. Then we have the following isomorphism for each $p$: $H^p(X,\mathcal{A}) \simeq (H^p(\tilde{X})\otimes \bar{A})^\pi$.
Whether the decomposition hypothesis actually holds may depend on the properties of the group $\pi$. For instance, it does hold if $\pi$ is compact (finite, in particular). Other cases, have to be examined individually.
Start with the short exact sequence of complexes
The corresponding long exact sequence in cohomology is equivalent to the short exact sequences
for each value of $p$. The reason that all the connecting maps in the long exact sequence are zero is representation theoretic, since all the relevant maps are $\pi$-equivariant. Since, $H^{p+1}(X,\mathcal{A})$ carries a trivial representation by $\pi$, while the $H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}, d_{\tilde{\nabla}})$ representation has no $\pi$-invariant subspace, by Schur's lemma, the only equivariant map from the latter to the former is zero. From the same observation, we easily see that the inclusion of $H^p(X,\mathcal{A})$ must coincide with the $\pi$-invariant subspace $(H^p(\tilde{X}) \otimes \bar{A})^\pi$.
The presentation of cohomology of $X$ with local coefficients $\mathcal{A}$ as $\pi$-invariant de Rham cohomology of the universal covering space $\tilde{X}$ twisted by the holonomy representation on the stalk $\bar{A}$ is originally due to (Eilenberg 47). It is also discussed in Chapter VI of (Whitehead 78). The idea to look at the $\pi$-invariant subspace of the twisted de Rham cohomology of the universal covering space scan be found in an answer by Peter Michor on MathOverlflow.
The above result can be seen as an effective way to compute the sheaf cohomology groups $H^p(X,\mathcal{A})$ since all it requires is the knowledge of the following finite dimensional representations of the fundamental group $\pi = \pi_1(X)$: the deck transformations on the de Rham cohomology $H^k(\tilde{X})$ of the covering space, and the holonomy representation on a typical stalk $\bar{A}$ of the locally constant sheaf $\mathcal{A}$. Obtaining the invariant subspace of their tensor product can then be done using usual representation theory methods, which involve only finite dimensional linear algebra. Unfortunately, it appears that the requirement that $\pi$ is finite is rather important for the argument. It is not entirely clearly how to proceed if, for instance $\pi = \mathbb{Z}$ or is non-abelian and infinite.
twisted ordinary cohomology, twisted K-theory, twisted tmf?
A discussion of $\pi_1(X)$-twisted ordinary cohomology is in
For the special case of generalized (Eilenberg–Steenrod) cohomology twisted by a $G$-principal bundle see section 22.1 of
This in turn is based on the definition of twisted K-theory given in
Details on Larmore’s work on twisted cohomology are at
The abstract discussion of twisted nonabelian cohomology in $\infty$-toposes is in
The abstract discussion of twisted ES-type cohomology in the stable (infinity,1)-category of spectra is in
Matthew Ando, Andrew Blumberg, David Gepner, Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and $C^*$-algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)
Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology (arXiv:1403.4325)
The presentation of cohomology with local coefficients in terms $\pi_1$-equivariant de Rham cohomology on the universal covering space is discussed in
The oldest meaning of twisted cohomology is that of cohomology with local coefficients (see above).
For more on the history of that notion see
In the following we shall abbreviate
Searching MathSciNet for twisted cohomology led to the following chronology: It missed older references in which the phrase was not used but the concept was in the sense of local coefficient systems – ancient and honorable.
Most notably missing are
Topologie der Komplexe_, Mathematik und ihre Anwendungen in Physik und Technik,_(But note that reprints appear, sans reviews. There is a short English and longer German review on Zentralblatt)
Norman Steenrod (1942,1943)
Olum (thesis 1947, published 1950)
Next come several that involve twisted differentials more generally.
Few are in terms of homotopy of spaces
tc ops should be treated as a single phrase – it may be that the ops are twisted, not the cohomology
1966 McClendon thesis – summarized in
1967 Emery Thomas tc ops
1967 Larmore tc ops
1969 McClendon tc ops
1969 Larmore tc
1970 Peterson tc ops
1971 McClendon tc ops
1972 Deligne Weil conjecture for K3 tc – meaning?
1972 Larmore tc
1973 Larmore and Thomas tc
1973 Larmore tc
gap
1980 Coelho & Pesennec tc
1980 Tsukiyama sequel to McClendon
1983 Coelho & Pesennec tc
1985 Morava but getsted at 1975 ??
1986 Fried tc
1988 Baum & Connes ??
1989 Lott torsion
1990 Dwork ??
1993 Gomez–Tato tc minimal models
1993 Duflo & Vergne diff tc
1993 Vaisman tc and connections
1993 Mimachi tc and holomorphic
1994 Kita tc and intersection
1995 Cho, Mimachi and Yoshida tc and configs
1995 Cho, Mimachi tc and intersection
1996 Iwaski and Kita tc de rham
1996 Asada nc geom and strings
1997 H Kimura tc de Rham and hypergeom
1998 Michael Farber, Gabriel Katz?, Jerome Levine?, Morse theory of harmonic forms, Topology, (Volume 37, Issue 3, May 1998, Pages 469–483)
1998 Knudson tc SL_n
1998 Morita tc de Rham
1999 Kachi, Mtsumoto, Mihara tc and intersection
1999 Hanamura & Yoshida Hodge tc
1999 Felshtyn & Sanchez–Morgado Reidemeister torsion
1999 Haraoka hypergeom
2000 Tsou & Zois tc de rham
2000 Manea tc Czech
2001 Royo Prieto tc Euler
2001 Takeyama q-twisted
2001 Gaberdiel &Schaefr–Nameki tc of Klein bottle
2001 Iwaskai tc deRham
2001 Proc Rims tc and DEs and several papers in this book
2001 Knudson tc SL_n
2001 Royo Prieto tc as $d+k\wedge$
2001 Barlewtta & Dragomir tc and integrability
2002 Lueck $L^2$
2002 Verbitsky HyperKahler, torsion, etc
2003 Etingof & Grana tc of Carter, Elhamdadi and Saito
2003 Cruikshank tc of Eilenberg
2003 various in Proc NATO workshop
2003 Dimca tc of hyperplanes
2004 Kirk & Lesch tc and index
2004 Bouwknegt, Evslin, Mathai tc and tK
2004 Bouwknegt, Hannbuss, Mathai tc in re: T-duality
2005 Bouwknegt, Hannbuss, Mathai tc in re: T-duality
2005 Bunke & Schick tc in re: T-duality
2006 Dubois tc and Reidemeister (elsewhere he considers twisted Reidemesiter)
2006 Bunke & Schick tc in re: T-duality
2006 Sati
2006 Atiyah & Segal tc and tK
2007 Mickelsson & Pellonpaa tc and tK
2007 Sugiyama in re: Galois and Reidemeister
2007 Bunke, Schick, Spitzweck tc in re: gerbes
2008 Kawahara hypersurfaces