We define categories of stratified manifolds (s-manifolds) and stratified manifolds with corners (s-manifolds with corners). An s-manifold ${\boldsymbol{X}}$ of dimension n is a Hausdorff, locally compact topological space X with a stratification $X=\coprod _{i\in I}X<^>i$ into locally closed subsets $X<^>i$ which are smooth manifolds of dimension $\leqslant n$, satisfying some conditions. S-manifolds can be very singular, but still share many good properties with ordinary manifolds, for example, an oriented s-manifold ${\boldsymbol{X}}$ has a fundamental class $[{\boldsymbol{X}}]_{\rm fund}$ in Steenrod homology $H_n<^>{\rm St}(X,{\mathbin {\mathbb {Z}}})$, and transverse fibre products exist in the category of s-manifolds. S-manifolds are designed for applications in Symplectic Geometry. In future work we hope to show that after suitable perturbations, the moduli spaces ${\mathbin {\smash{\, \, \overline{\!\!\mathcal {M}\!}\, }}\vphantom{\cal M}}$ of J-holomorphic curves used to define Gromov-Witten invariants, Lagrangian Floer cohomology, Fukaya categories, and so on, can be made into s-manifolds or s-manifolds with corners, and their fundamental classes used to define Gromov-Witten invariants, Lagrangian Floer cohomology.

We generalize complex manifolds to manifolds with corners X, and to manifolds with generalized corners (g-corners) in the sense of the second author arXiv:1501.00401, using complex structures on the b-tangent bundle (log tangent bundle) ^bTX. We prove a formal Newlander-Nirenberg type theorem showing that along each corner stratum of X, the b-complex structure agrees with a standard model to infinite order. In the sequel we show that if S is a log smooth log ℂ-scheme, or log smooth log complex analytic space, then the Kato-Nakayama space S^ KN has the structure of a b-complex manifold with g-corners. Using our Newlander-Nirenberg theorem we give necessary and sufficient conditions for a b-complex manifold with g-corners to be a Kato-Nakayama space.

To define enumerative invariants in geometry, one often needs orientations on moduli spaces of geometric objects. This monograph develops a new bordism-theoretic point of view on orientations of moduli spaces. Let X be a manifold with geometric structure, and M a moduli space of geometric objects on X. Our theory aims to answer the questions: (i) Can we prove M is orientable for all X, M? (ii) If not, can we give computable sufficient conditions on X that guarantee M is orientable? (iii) Can we specify extra data on X which allow us to construct a canonical orientation on M? We define 'bordism categories', such as Bord_n^Spin(BG) with objects (X,P) for X a compact spin n-manifold and P→ X a principal G-bundle, for G a Lie group. Bordism categories can be understood by computing bordism groups of classifying spaces using Algebraic Topology. Orientation problems are encoded in functors from a bordism category to ℤ_2-torsors. We apply our theory to study orientability and canonical orientations for moduli spaces of G_2-instantons and associative 3-folds in G_2-manifolds, for moduli spaces of Spin(7)-instantons and Cayley 4-folds in Spin(7)-manifolds, and for moduli spaces of coherent sheaves on Calabi-Yau 4-folds. The latter are needed to define Donaldson-Thomas type invariants of Calabi-Yau 4-folds. In many cases we prove orientability of M, and show canonical orientations can be defined using a 'flag structure'.

Schemes in algebraic geometry can have singular points, whereas differential geometers typically focus on manifolds which are nonsingular. However, there is a class of schemes, 'C∞-schemes', which allow differential geometers to study a huge range of singular spaces, including 'infinitesimals' and infinite-dimensional spaces. These are applied in synthetic differential geometry, and derived differential geometry, the study of 'derived manifolds'. Differential geometers also study manifolds with corners. The cube is a 3-dimensional manifold with corners, with boundary the six square faces. This book introduces 'C∞-schemes with corners', singular spaces in differential geometry with good notions of boundary and corners. They can be used to define 'derived manifolds with corners' and 'derived orbifolds with corners'. These have applications to major areas of symplectic geometry involving moduli spaces of J-holomorphic curves. This work will be a welcome source of information and inspiration for graduate students and researchers working in differential or algebraic geometry.

Suppose $(X, g)$ is a compact, spin Riemannian 7-manifold, with Dirac operator $D$. Let $G$ be SU$(m)$ or U$(m)$, and $E\to X$ be a rank $m$ complex bundle with $G$-structure. Write ${\mathcal B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal ${\mathbb Z}_2$-bundle $O^D_E\to{\mathcal B}_E$ parametrizing orientations of det$\,D_{{\rm Ad }A}$ for twisted elliptic operators $D_{{\rm Ad }A}$ at each $[A]$ in ${\mathcal B}_E$. A theorem of Walpuski shows $O^D_E$ is trivializable. We prove that if we choose an orientation for det$\,D$, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of $O^D_E$ for all such bundles $E\to X$, satisfying natural compatibilities. Now let $(X,\varphi,g)$ be a compact $G_2$-manifold, with d$(*\varphi)=0$. Then we can consider moduli spaces ${\mathcal M}_E^{G_2}$ of $G_2$-instantons on $E\to X$, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with ${\mathcal M}_E^{G_2}\subset{\mathcal B}_E$. The restriction of $O^D_E$ to ${\mathcal M}_E^{G_2}$ is the ${\mathbb Z}_2$-bundle of orientations on ${\mathcal M}_E^{G_2}$. Thus, our theorem induces canonical orientations on all such $G_2$-instanton moduli spaces ${\mathcal M}_E^{G_2}$. This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of $G_2$-manifolds $(X,\varphi,g)$ by counting moduli spaces ${\mathcal M}_E^{G_2}$, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6. A third paper Cao-Gross-Joyce arXiv:1811.09658 studies orientations on moduli spaces in dimension 8.

This is the second paper of a series that develops a bordism-theoretic point of view on orientations in enumerative geometry. The first paper is arXiv:2312.06818. This paper focuses on those applications to gauge theory that can be established purely using formal arguments and calculations from algebraic topology. We prove that the orientability of moduli spaces of connections in gauge theory for all principal G-bundles P→ X over compact spin n-manifolds at once is equivalent to the vanishing of a certain morphism Ω_n^ Spin(ℒ BG)→ℤ_2 on the n-dimensional spin bordism group of the free loop space of the classifying space of G, and we give a complete list of all compact, connected Lie groups G for which this holds. Moreover, we apply bordism techniques to prove that mod-8 Floer gradings exist for moduli spaces of G_2-instantons for all principal SU(2)-bundles. We also prove that there are canonical orientations for all principal U(m)-bundles P→ X over compact spin 8-manifolds satisfying c_2(P)-c_1(P)^2=0. The proof is based on an interesting relationship to principal E_8-bundles. These canonical orientations play an important role in many conjectures about Donaldson-Thomas type invariants on Calabi-Yau 4-folds, and resolve an apparent paradox in these conjectures.

Let $X$ be a compact Calabi-Yau 3-fold, and write $\mathcal M,\bar{\mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$, analogues of canonical bundles. Orientation data on $\mathcal M,\bar{\mathcal M}$ is an isomorphism class of square root line bundles $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman arXiv:1006.270 in their theory of motivic Donaldson-Thomas invariants, and is important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds $X$ with a spin smooth projective compactification $X\hookrightarrow Y$. This proves a long-standing conjecture in Donaldson-Thomas theory. These are special cases of a more general result. Let $X$ be a spin smooth projective 3-fold. Using the spin structure we construct line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$. We define spin structures on $\mathcal M,\bar{\mathcal M}$ to be isomorphism classes of square roots $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$. We prove that natural spin structures exist on $\mathcal M,\bar{\mathcal M}$. They are equivalent to orientation data when $X$ is a Calabi-Yau 3-fold with the trivial spin structure. We prove this using our previous paper arXiv:1908.03524, which constructs 'spin structures' (square roots of a certain complex line bundle $K_P\to\mathcal B_P$) on differential-geometric moduli stacks $\mathcal B_P$ of connections on a principal U$(m)$-bundle $P\to X$ over a compact spin 6-manifold $X$.

Enumerative invariants in Algebraic Geometry 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=a$ in some geometric problem, using a virtual class $[{\cal M}_a^{\rm ss}(\tau)]_{\rm virt}$ in homology, for the moduli spaces ${\cal M}_a^{\rm st}(\tau)\subseteq{\cal M}_a^{\rm ss}(\tau)$ of $\tau$-(semi)stable objects. We get numbers by taking integrals $\int_{[{\cal M}_a^{\rm ss}(\tau)]_{\rm virt}}\Upsilon$ for cohomology classes $\Upsilon$. Let $\cal A$ be a $\mathbb C$-linear abelian category in Algebraic Geometry. There are two moduli stacks of objects in $\cal A$: the usual moduli stack $\cal M$, and the 'projective linear' moduli stack $\cal M^{\rm pl}$. We give $H_*({\cal M})$ the structure of a vertex algebra, and $H_*({\cal M}^{\rm pl})$ a Lie algebra. Virtual classes $[{\cal M}_a^{\rm ss}(\tau)]_{\rm virt}$ lie in $H_*({\cal M}^{\rm pl})$. We develop a universal theory of enumerative invariants in such $\mathcal A$. Virtual classes $[{\cal M}_a^{\rm ss}(\tau)]_{\rm virt}$ are only defined when ${\cal M}_a^{\rm st}(\tau)={\cal M}_a^{\rm ss}(\tau)$. We define invariants $[{\cal M}_a^{\rm ss}(\tau)]_{\rm inv}$ in $H_*({\cal M}^{\rm pl})$ for all $a$, with $[{\cal M}_a^{\rm ss}(\tau)]_{\rm inv}=[{\cal M}_a^{\rm ss}(\tau)]_{\rm virt}$ when ${\cal M}_a^{\rm st}(\tau)={\cal M}_a^{\rm ss}(\tau)$. If $\tau,\tau'$ are stability conditions on $\cal A$, we prove a wall-crossing formula writing $[{\cal M}_a^{\rm ss}(\tau')]_{\rm inv}$ in terms of the $[{\cal M}_b^{\rm ss}(\tau)]_{\rm inv}$, using the Lie bracket on $H_*({\cal M}^{\rm pl})$. We apply our results for $\cal A$ the representations of a quiver or quiver with relations, or coh$(X)$ for $X$ a curve, surface or Fano 3-fold, or a category of 'pairs' in coh$(X)$ for $X$ a curve or surface. This proves conjectures in Gross-Joyce-Tanaka arXiv:2005.05637.

Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $P\to X$ a principal $G$-bundle, and ${\mathcal B}_P$ the infinite-dimensional moduli space of all connections $\nabla_P$ on $P$ modulo gauge, as a topological stack. For each $[\nabla_P]\in{\mathcal B}_P$, we can consider the twisted elliptic operator $D^{\nabla_{Ad(P)}}$ on X. This is a continuous family of elliptic operators over the base ${\mathcal B}_P$, and so has an orientation bundle $O^D_P\to{\mathcal B}_P$, a principal ${\mathbb Z}_2$-bundle parametrizing orientations of Ker$D^{\nabla_{Ad(P)}}\oplus$Coker$D^{\nabla_{Ad(P)}}$ at each $[\nabla_P]$. An orientation on $({\mathcal B}_P,D)$ is a trivialization $O^D_P\cong{\mathcal B}_P\times{\mathbb Z}_2$. In gauge theory one studies moduli spaces $\mathcal M$ of connections $\nabla_P$ on $P$ satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds $(X, g)$. Under good conditions $\mathcal M$ is a smooth manifold, and orientations on $({\mathcal B}_P,D)$ pull back to orientations on $\mathcal M$ in the usual sense under the inclusion ${\mathcal M}\hookrightarrow{\mathcal B}_P$. This is important in areas such as Donaldson theory, where one needs an orientation on $\mathcal M$ to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on $({\mathcal B}_P,D)$, after fixing some algebro-topological information on $X$. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, Kapustin-Witten and Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds. Two sequels arXiv:1811.02405, arXiv:1811.09658 discuss orientations in 7 and 8 dimensions.

There is a strong analogy between compact, torsion-free G(2)-manifolds (X, phi, phi) and Calabi Yau 3-folds (Y, J, g, omega). We can also generalize (X, phi, phi) to 'tamed almost G2-manifolds' (X, phi, psi), where we compare phi with omega and psi with J. Associative 3-folds in X, a special kind of minimal submanifold, are analogous to J-holomorphic curves in Y. Several areas of Symplectic Geometry Gromov Witten theory, Quantum Cohomology, L'agrangian Floer cohomology, Fukaya categories are built using `counts' of moduli spaces of J-holomorphic curves in Y, but give an answer depending only on the symplectic manifold (Y, omega), not on the (almost) complex structure J. We investigate whether it may be possible to define interesting invariants of tamed almost G(2)-manifolds (X, phi, psi) by 'counting' compact associative 3-folds N subset of X, such that the invariants depend only on phi, and are independent of the 4-form psi used to define associative 3-folds. We conjecture that one can define a superpotential Phi(psi) : U -> Lambda>0 'counting' associative Q-homology 3-spheres N C X which is deformation-invariant in psi for phi fixed, up to certain reparametrizations T : U -> U of the base U = Hom(H-3(X; Z), 1 + A(>0))), where A(>0) is a Novikov ring. Using this we define a notion of `G(2) quantum cohomology'. We also discuss Donaldson and Segal's proposal from their 2011 work to define invariants 'counting' G(2)-instantons on tamed almost G(2)-manifolds, with 'compensation terms' counting weighted pairs of a G(2)-instanton and an associative 3-fold, and suggest some modifications to it.

Manifolds with boundary and with corners form categories Man⊂ Man^b⊂ Man^c. A manifold with corners X has two notions of tangent bundle: the tangent bundle TX, and the b-tangent bundle ^bTX. The usual definition of smooth structure uses TX, as f:X→ℝ is defined to be smooth if ∇^kf exists as a continuous section of ⊗^kT^*X for all k≥ 0. We define 'manifolds with analytic corners', or 'manifolds with a-corners', with a different smooth structure, in which roughly f:X→ℝ is smooth if ^b∇^kf exists as a continuous section of ⊗^k(^bT^*X) for all k≥ 0. These are different from manifolds with corners even when X=[0,∞), for instance x^α:[0,∞)→ℝ is smooth for all real α≥ 0 when [0,∞) has a-corners. Manifolds with a-boundary and with a-corners form categories Man⊂ Man^ab⊂ Man^ac, with well behaved differential geometry. Partial differential equations on manifolds with boundary may have boundary conditions of two kinds: (i) 'at finite distance', e.g. Dirichlet or Neumann boundary conditions, or (ii) 'at infinity', prescribing the asymptotic behaviour of the solution. We argue that manifolds with corners should be used for (i), and with a-corners for (ii). We discuss many applications of manifolds with a-corners in boundary problems of type (ii), and to singular p.d.e. problems involving 'bubbling', 'neck-stretching' and 'gluing'.

We prove two main results. (1) Suppose that L is a closed, embedded, exact special Lagrangian m-fold in C-m asymptotic at infinity to the union Pi(1) boolean OR Pi(2) of two transverse special Lagrangian planes Pi(1,) Pi(2) in C-m for m >= 3. Then L is one of the explicit Lawlor neck family of examples found by Lawlor. (2) Suppose that L is a closed, embedded, exact Lagrangian mean curvature flow expander in Cm asymptotic at infinity to the union Pi(1) boolean OR Pi(2) of two transverse Lagrangian planes Pi(1), Pi(2) in C-m for m >= 3. Then L is one of the explicit family of examples in recent work by Joyce, Lee, and Tsui. If instead L is immersed rather than embedded, the only extra possibility in (1), (2) is L = Pi(1), Pi(2). Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve J-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology.

This is a survey of the author's paper arXiv:1409.6908 and in-progress book. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli spaces of J-holomorphic curves. We propose a new definition of Kuranishi space, which has the nice property that they form a 2-category Kur. Thus the homotopy category Ho( Kur) is an ordinary category of Kuranishi spaces. Any Fukaya-Oh-Ohta-Ono (FOOO) Kuranishi space X can be made into a compact Kuranishi space X' uniquely up to equivalence in Kur (that is, up to isomorphism in Ho( Kur)), and conversely any compact Kuranishi space X' comes from some (nonunique) FOOO Kuranishi space X. So FOOO Kuranishi spaces are equivalent to ours at one level, but our definition has better categorical properties. The same holds for McDuff and Wehrheim's 'Kuranishi atlases' in arXiv:1508.01556. Using results of Yang on polyfolds and Kuranishi spaces surveyed in arXiv:1510.06849, a compact topological space X with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185) can be made into a Kuranishi space X uniquely up to equivalence in Kur. Our Kuranishi spaces are based on the author's theory of Derived Differential Geometry (see e.g. arXiv:1206.4207), the study of classes of derived manifolds and orbifolds that we call 'd-manifolds' and 'd-orbifolds'. There is an equivalence of 2-categories Kur≃ dOrb, where dOrb is the 2-category of d-orbifolds. So Kuranishi spaces are really a form of derived orbifold. We discuss the differential geometry of Kuranishi spaces.

Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of "stability" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in[0,\infty)\}$ with $L^0=L$ should exist for all time, and $L^\infty=\lim_{t\to\infty}L^t$ should be the unique special Lagrangian in the Hamiltonian isotopy class of $L$. This paper is an attempt to update the Thomas-Yau conjectures, and discuss related issues. It is a folklore conjecture that there exists a Bridgeland stability condition $(Z,\mathcal P)$ on the derived Fukaya category $D^b\mathcal F(M)$ of $M$, such that an isomorphism class in $D^b\mathcal F(M)$ is $(Z,\mathcal P)$-semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique. We conjecture that if $(L,E,b)$ is an object in an enlarged version of $D^b\mathcal F(M)$, where $L$ is a compact, graded Lagrangian in $M$ (possibly immersed, or with "stable singularities"), $E\to M$ a rank one local system, and $b$ a bounding cochain for $(L,E)$ in Lagrangian Floer cohomology, then there is a unique family $\{(L^t,E^t,b^t):t\in[0,\infty)\}$ such that $(L^0,E^0,b^0)=(L,E,b)$, and $(L^t,E^t,b^t)\cong(L,E,b)$ in $D^b\mathcal F(M)$ for all $t$, and $\{L^t:t\in[0,\infty)\}$ satisfies Lagrangian MCF with surgeries at singular times $T_1,T_2,\dots,$ and in graded Lagrangian integral currents we have $\lim_{t\to\infty}L^t=L_1+\cdots+L_n$, where $L_j$ is a special Lagrangian integral current of phase $e^{i\pi\phi_j}$ for $\phi_1>\cdots>\phi_n$, and $(L_1,\phi_1),\ldots,(L_n,\phi_n)$ correspond to the decomposition of $(L,E,b)$ into $(Z,\mathcal P)$-semistable objects. We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_1,T_2,\ldots.$

For each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex $(MC_*(Y;R),\partial)$ is generated by quadruples $[V,n,s,t]$ satisfying relations, where $V$ is an oriented manifold with corners, $n\in\mathbb N$, and $s:V\to{\mathbb R}^n$, $t:V\to Y$ are smooth with $s$ proper near 0 in ${\mathbb R}^n$. We show that $MH_*(Y;R),MH^*(Y;R)$ satisfy the Eilenberg-Steenrod axioms, and so are canonically isomorphic to conventional (co)homology. The usual operations on (co)homology -- pushforwards $f_*$, pullbacks $f^*$, fundamental classes $[Y]$ for compact oriented $Y$, cup, cap and cross products $\cup,\cap,\times$ -- are all defined and well-behaved at the (co)chain level. Chains $MC_*(Y;R)$ form flabby cosheaves on $Y$, and cochains $MC^*(Y;R)$ form soft sheaves on $Y$, so they have good gluing properties. We also define $compactly$-$supported$ $M$-$cohomology$ $MH^*_{cs}(Y;R)$, $locally$ $finite$ $M$-$homology$ $MH_*^{lf}(Y;R)$ (a kind of Borel-Moore homology), and two variations on the entire theory, $rational$ $M$-($co$)$homology$ and $de$ $Rham$ $M$-($co$)$homology$. All of these are canonically isomorphic to the corresponding type of conventional (co)homology. The reason for doing this is that our M-(co)homology theories are very well behaved at the (co)chain level, and will be better than other (co)homology theories for some purposes, particularly in problems involving transversality. In a sequel we will construct virtual classes and virtual chains for Kuranishi spaces in M-(co)homology, with a view to applications of M-(co)homology in areas of Symplectic Geometry involving moduli spaces of $J$-holomorphic curves.

This is a survey of [20]. We introduce a 2-category dMan of d-manifolds, new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived manifolds' of Spivak [30]. Manifolds Man embed in dMan as a full (2-) subcategory. There are also 2-categories dMan(b), dMan(c) of d-manifolds with boundary and with corners, and orbifold versions of these dOrb, dOrb(b), dOrb(c), d-orbifolds.Much of differential geometry extends very nicely to d-manifolds - immersions, submersions, submanifolds, transverse fibre products, orientations, etc. Compact oriented d-manifolds have virtual classes.Many areas of symplectic geometry involve 'counting' moduli spaces (M) over bar (g),(m)(J, beta) of J-holomorphic curves to define invariants, Floer homology theories, etc. Such (M) over bar (g),(m)(J, beta) are given the structure of Kuranishi spaces in the work of Fukaya, Oh, Ohta and Ono [9], but there are problems with the theory. The author believes the 'correct' definition of Kuranishi spaces is that they are d-orbifolds with corners. D-manifolds and d-orbifolds will have applications in symplectic geometry, and elsewhere.For brevity, this survey focusses on d-manifolds without boundary. A longer and more detailed summary of the book is given in [21].

'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of $J$-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a satisfactory definition of Kuranishi space has been the subject of recent debate (see e.g. arXiv:1208.1340, arXiv:1209.4410, arXiv:1510.06849). We propose three new definitions of Kuranishi space: a simple 'manifold' version, '$\mu$-Kuranishi spaces', which form an ordinary category $\boldsymbol\mu\bf Kur$; a more complicated 'manifold' version, 'm-Kuranishi spaces', which form a weak 2-category $\bf mKur$; and an 'orbifold' version, 'Kuranishi spaces', which form a weak 2-category $\bf Kur$. These are related by an equivalence of categories $\boldsymbol\mu{\bf Kur}\simeq{\rm Ho}({\bf mKur})$, where ${\rm Ho}({\bf mKur})$ is the homotopy category of $\bf mKur$, and by a full and faithful embedding ${\bf mKur}\hookrightarrow\bf Kur$. We also define ($\mu$-, m-)Kuranishi spaces with boundary, and with corners. We hope our definitions will become accepted as final, replacing previous definitions. Any Fukaya-Oh-Ohta-Ono Kuranishi space $\bf X$ can be made into a compact Kuranishi space $\bf X'$ uniquely up to equivalence in $\bf Kur$ (that is, up to isomorphism in ${\rm Ho}({\bf Kur})$). The same holds for topological spaces with Fukaya-Oh-Ohta-Ono 'good coordinate systems', and for McDuff and Wehrheim's 'Kuranishi atlases' in arXiv:1508.01556. A compact topological space $\bf X$ with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder can be made into a Kuranishi space $\bf X$ uniquely up to equivalence in $\bf Kur$. This book is surveyed in arXiv:1510.07444.