Volume 25, Issue 4

In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book [B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups. Part 2. The Applications of (Rational) Homotopy Theory Methods, Math. Surveys Monogr. 217, American Mathematical Society, Providence, 2017]. In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite-type rational cohomology groups).

We show that for a field K and n≥1{n\geq 1}, the soberification 𝒮⁢(𝔸n⁢(K)){\mathcal{S}(\mathbb{A}^{n}(K))} of the affine n -space 𝔸n⁢(K){\mathbb{A}^{n}(K)} over K is homeomorphic to its spectralization ℬ⁢𝒮⁢(𝔸n⁢(K)){\mathcal{BS}(\mathbb{A}^{n}(K))}, and it can be embedded into the spectrum Spec⁢(K⁢[X1,…,Xn]){\mathrm{Spec}(K[X_{1},\dots,X_{n}])}. Moreover, if the field K is algebraically closed, then there are homeomorphisms 𝒮⁢(𝔸n⁢(K))≈ℬ⁢𝒮⁢(𝔸n⁢(K))≈Spec⁢(K⁢[X1,…,Xn])\mathcal{S}(\mathbb{A}^{n}(K))\approx\mathcal{BS}(\mathbb{A}^{n}(K))\approx% \mathrm{Spec}(K[X_{1},\dots,X_{n}]). We also show that for a space X , the subspace z⁢Spec⁢(C⁢(X))⊆Spec⁢(C⁢(X))z\mathrm{Spec}(C(X))\subseteq\mathrm{Spec}(C(X)) of prime z-ideals of the ring C⁢(X){C(X)} of real-valued continuous functions on X is homeomorphic to the space z⁢𝒮⁢ℛ⁢(X){z\mathcal{SR}(X)} of prime z-filters with an appropriate topology and there is a homeomorphism ℬ⁢𝒮⁢(X)≈z⁢Spec⁢(C⁢(X)){\mathcal{BS}(X)\approx z\mathrm{Spec}(C(X))} provided X is perfectly normal.

Given a map f:X→Y{f\colon X\to Y}, we extend Gottlieb’s result to the generalized Gottlieb group Gf⁢(Y,f⁢(x0)){G^{f}(Y,f(x_{0}))} and show that the canonical isomorphism π1⁢(Y,f⁢(x0))⁢→≈⁢𝒟⁢(Y){\pi_{1}(Y,f(x_{0}))\overset{\approx}{\to}\mathcal{D}(Y)} restricts to an isomorphism Gf⁢(Y,f⁢(x0))⁢→≈⁢𝒟f~0⁢(Y){G^{f}(Y,f(x_{0}))\overset{\approx}{\to}\mathcal{D}^{\tilde{f}_{0}}(Y)}, where 𝒟f~0⁢(Y){\mathcal{D}^{\tilde{f}_{0}}(Y)} is some subset of the group 𝒟⁢(Y){\mathcal{D}(Y)} of deck transformations of Y for a fixed lifting f~0{\tilde{f}_{0}} of f with respect to universal coverings of X and Y , respectively.

Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds.

Kadeishvili proposes a minimal C∞{C_{\infty}}-algebra as a rational homotopy model of a space. We discuss a cyclic version of this Kadeishvili C∞{C_{\infty}}-model and apply it to classifying rational Poincaré duality spaces. We classify 1-connected minimal cyclic C∞{C_{\infty}}-algebras whose cohomology algebras are those of (Sp×Sp+2⁢q-1)⁢♯⁢(Sq×S2⁢p+q-1){(S^{p}\times S^{p+2q-1})\sharp(S^{q}\times S^{2p+q-1})}, where 2≤p≤q{2\leq p\leq q}. We also include a proof of the decomposition theorem for cyclic A∞{A_{\infty}} and C∞{C_{\infty}}-algebras.

The notion of a cyclic ∞{\infty}-simplicial module is introduced. The homotopy invariance of the structure of a cyclic ∞{\infty}-simplicial module is proved. The conception of the cyclic homology of cyclic ∞{\infty}-simplicial modules is developed. For the cyclic homology of cyclic ∞{\infty}-simplicial modules, the analogue of the Connes–Tsygan exact sequence is obtained.

Let P be an n -gon with n≥3{n\geq 3}. There is a formal combinatorial A∞{A_{\infty}}-coalgebra structure on cellular chains C*⁢(P){C_{*}(P)} with non-vanishing higher order structure when n≥5{n\geq 5}. If Xg{X_{g}} is a closed compact surface of genus g≥2{g\geq 2} and Pg{P_{g}} is a polygonal decomposition, the quotient map q:Pg→Xg{q\colon P_{g}\to X_{g}} projects the formal A∞{A_{\infty}}-coalgebra structure on C*⁢(Pg){C_{*}(P_{g})} to a quotient structure on C*⁢(Xg){C_{*}(X_{g})}, which persists to homology H∗⁢(Xg;ℤ2){H_{\ast}(X_{g};\mathbb{Z}_{2})}, whose operations are determined by the quotient map q , and whose higher order structure is non-trivial if and only if Xg{X_{g}} is orientable with g≥2{g\geq 2} or unorientable with g≥3{g\geq 3}. But whether or not the A∞{A_{\infty}}-coalgebra structure on homology observed here is topologically invariant is an open question.

We propose in this article a global understanding of, on the one hand, the homological perturbation theorem (HPT) and, on the other hand, of Robin Forman’s theorems about the discrete vector fields (DVFs). Forman’s theorems become a simple and clear consequence of the HPT. Above both subjects, the homological hexagonal lemma quite elementary.

A decade ago, I wrote a tribute to Tornike Kadeishvili in honor of his 60th birthday [J. Stasheff, A twisted tale of cochains and connections, Georgian Math. J. 17 2010, 1, 203–215]. Time marches on! Now for his 70th I will update (with some repetitions) that tribute, also to the Georgian school.

Based on Kadeishvili’s original theorem inducing A∞{A_{\infty}}-algebra structures on the homology of dg-algebras, several directions of algorithmic research in A∞{A_{\infty}}-algebras have been pursued. In this paper, we survey the work done on calculating explicit A∞{A_{\infty}}-algebra structures from homotopy retractions, in group cohomology and in persistent homology.