# nLab differential topology

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Differential topology is the subject devoted to the study of algebro-topological and homotopy-theoretic properties of differentiable manifolds, smooth manifolds and related differential geometric spaces such as stratifolds, orbifolds and more generally differentiable stacks.

A key part of differential topology is cobordism theory, where the Pontryagin-Thom theorem relates the stable homotopy theory of Thom spectra to cobordism classes of smooth (sub-)manifolds (for instance cohomotopy to normally framed cobordism).

Differential topology is also concerned with the problem of finding out which topological (or PL) manifolds allow a differentiable structure and the degree of nonuniqueness of that structure if they do (e.g. exotic smooth structures). It is also concerned with concrete constructions of (co)homology classes (e.g. characteristic classes) for differentiable manifolds and of differential refinements of cohomology theories.

More recently, the smooth Oka principle reveals a deep structure in differential topology which is visible in the full generality of higher differential geometry (smooth $\infty$-stacks).

## Examples

Many considerations, and classification problems, depend crucially on dimension, and the case of high-dimensional manifolds (the notion of ‘high’ depends on the problem) is often very different from the situation in each of the low dimensions; thus there are specialists’ subjects like $3$-(dimensional) topology and $4$-topology. There are restrictions on an underlying topology which is allowed for some sorts of additional structures on a differentiable manifold.

For example, only some even-dimensional differentiable manifolds allow for symplectic structure and only some odd-dimensional one allow for a contact structure; in these cases moreover special constructions of topological invariants like Floer homology and symplectic field theory exist.

This yields the relatively young subjects of symplectic and contact topologies, with the first significant results coming from Gromov. Any (Hausdorff paracompact finite-dimensional) differentiable manifold allows for riemannian structure however; therefore there is no special subject of ‘riemannian topology’.

## References

Though some of the basic results, methods and conjectures of differential topology go back to Poincaré, Whitney, Morse and Pontrjagin, it became an independent field only in the late 1950s and early 1960s with the seminal works of Smale, Thom, Milnor and Hirsch. Soon after the initial effort on foundations, mainly in the American school, a strong activity started in Soviet Union (Albert Schwarz, A. S. Mishchenko, S. Novikov, V. A. Rokhlin, M. Gromov…).

Introductions and monographs:

Survey with connections to algebraic topology: