# nLab cobordism hypothesis

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

The Cobordism Hypothesis states, roughly, that the (∞,n)-category of cobordisms $Bord_n^{fr}$ is the free symmetric monoidal (∞,n)-category with duals on a single object.

Since a fully extended topological quantum field theory may be identified with an monoidal (∞,n)-functor $Z : Bord_n \to C$, this implies that all these TQFTs are entirely determined by their value on the point: “the n-vector space of states” of the theory.

As motivation, notice that by Galatius-Tillmann-Madsen- 09Weiss we have that the loop space of the geometric realization of the framed cobordism category is equivalent to the sphere spectrum

$\Omega \Vert Cob_d^{fr} \Vert \simeq \lim_{\to_{n \to \infty}} Maps_*(S^n, S^n) \simeq \Omega^\infty S^\infty$

which can be understood as the free infinite loop space on the point.

## Formalization

In (Lurie) a formalization and proof of the cobordism hypothesis is described.

### For framed cobordisms

#### Statement

For $\mathcal{C}$ a symmetric monoidal (∞,n)-category write $Core(\mathcal{C})$ for its core (the maximal ∞-groupoid in $\mathcal{C}$).

For $\mathcal{C}$, $\mathcal{D}$ two symmetric monoidal (∞,n)-categories, write $Fun^\otimes(\mathcal{D}, \mathcal{C} )$ for the (∞,n)-category of symmetric monoidal (∞,n)-functors between them.

###### Definition

Write $Bord_n^{fr}$ be the symmetric monoidal (∞,n)-category of cobordisms with n-framing.

###### Theorem (cobordism hypothesis, framed version)

Evaluation of any such functor $F$ on the point ${*}$

$F \mapsto F({*})$

induces an (∞,n)-functor

$pt^* : Fun^\otimes(Bord_n^{fr} , \mathcal{C} ) \to \mathcal{C} .$

such that

• this factors through the core of $\mathcal{C}$;

• the map

$pt^* : Fun^\otimes(Bord_n^{fr} , \mathcal{C} ) \to Core(\mathcal{C})$

is an equivalence of (∞,n)-categories.

This is (Lurie, theorem 2.4.6).

###### Proof

The proof is based on

1. the Galatius-Madsen-Tillmann-Weiss theorem, which characterizes the geometric realization $|Bord_n^{or}|$ in terms of the suspension of the Thom spectrum;

2. Igusa’s connectivity result which he uses to show that putting “framed Morse functions” on cobordisms doesn’t change their homotopy type (theorem 3.4.7, page 73)

In fact, the Galatius-Madsen-Weiss theorem is now supposed to be a corollary of Lurie’s result.

#### Implications – The canonical $O(n)$-∞-action

One of the striking consequences of theorem 1 is that it implies that

###### Corollary

Every ∞-groupoid

$\mathcal{C}^{fd} \hookrightarrow \mathcal{C}$

of fully dualizable objects in a symmetric monoidal (∞,n)-category $\mathcal{C}$ carries a canonical ∞-action of (the ∞-group structure on the homotopy type of) the orthogonal group $O(n)$, induced by the action of $O(n)$ on the n-framing of the point in $Bord_n^{fr}$.

###### Example

The action in corollary 1 is

• for $n = 1$: the $O(1) = \mathbb{Z}/2\mathbb{Z}$ action action given by passing to dual objects;

• for $n = 2$ the $O(2)$-action the Serre automorphism.

• for $n = \infty$ the $O(n)$-action on n-fold loop spaces (see e.g. Gaudens-Menichi 07, section 5) (see also at orthogonal spectra).

###### Proposition

For all $n \in \mathbb{N}$, the canonical $SO$-∞-action on

$B^n \mathbb{Z} \in Ab_\infty(\infty Grpd) \hookrightarrow (\infty,n)CatWithDuals$

is trivial.

###### Proof

The action on a connective spectrum $\Omega^\infty X$ factors through the J-homomorphism

$SO \times \Omega^\infty X \stackrel{(J,id)}{\longrightarrow} \Omega^\infty S^\infty \times \Omega^\infty X \stackrel{precomp}{\longrightarrow} \Omega^\infty X \,.$

But on homotopy groups the image of J is pure torsion which means that for $\Omega^\infty X = B^n \mathbb{Z}$ the induced actions on homotopy groups are all trivial. From this and using the long exact sequence of homotopy groups it follows that the $\infty$-action itself is trivial.

### For cobordisms with extra topological structure

We discuss the cobordism hypothesis for cobordisms that are equipped with the extra structure of maps into some topological space equipped with a vector bundle. This is the case for which an extended TQFT is (the local refinement of) what has also been called an HQFT.

#### For cobordisms with any structure (“$(X,\xi)$-structure”)

###### Definition

Let $X$ be a topological space and $\xi \to X$ a real vector bundle on $X$ of rank $n$. Let $N$ be a smooth manifold of dimension $m \leq n$. An $(X,\xi)$-structure on $N$ consists of the following data

• A continuous function $f : N \to X$;

• $T N \oplus \mathbb{R}^{n-m} \simeq f^* \xi$

between the fiberwise direct sum of the tangent bundle $T N$ with the trivial rank $(n-m)$ bundle and the pullback of $\xi$ along $f$.

This is (Lurie, notation 2.4.16).

The two extreme cases of def. 2 are the following

###### Example

For $X = \ast$ the point and $\xi = \mathbb{R}^n$, then an $(X,\xi)$-structure is the same as an $n$-framing, hence

$(Bord_n^{(\ast, \mathbb{R}^n)}) \simeq Bord_n^{fr}$

reproduces the $(\infty,n)$-category of framed cobordisms of def. 1.

###### Example

For $X = B O(n)$ the classifying space of real vector bundles of rank $n$ (the delooping of the ∞-group $O(n)$ underlying the orthogonal group) and for $\xi = E O(n) \underset{O(n)}{\times} \mathbb{R}^n$ the vector bundle associated to the $O(n)$-universal bundle, then $(X,\xi)$-structure on $n$-dimensional manifolds is essentially no-structure (the maximal compact subgroup-inclusion $O(n)\to GL(n)$ is a weak homotopy equivalence). Cobordisms with this structure will also be called unoriented cobordisms

$Bord_n^{un} \coloneqq Bord_n^{(B O(n), E O(n)\underset{O(n)}{\times} \mathbb{R}^n)} \,.$

Accordingly, for $X = B SO(n)$ the delooping of the special orthogonal group, the corresponding $(X,\xi)$-structure makes oriented manifolds

$Bord_n^{or} \coloneqq Bord_n^{(B SO(n), E SO(n)\underset{SO(n)}{\times} \mathbb{R}^n)} \,.$

Generally:

###### Example

For $\chi \colon G \to O(n)$ a topological group mapping via a homomorphism to $O(n)$, then $X = B G$ and $\xi = \chi^\ast (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)$, the $(X,\xi)$-structure is G-structure.

This we get to below.

###### Definition

Let $X$ be a topological space and $\xi \to X$ an $n$-dimensional vector bundle. The (∞,n)-category $Bord_n^{(X, \xi)}$ is defined analogously to $Bord_n$ but with all manifolds equipped with $(X,\xi)$-structure, def. 2.

This is (Lurie, def. 2.4.17).

###### Theorem

Let $\mathcal{C}$ be a symmetric monoidal (∞,n)-category with duals, let $X$ be a CW-complex, let $\xi \to X$ be an $n$-dimensional vector bundle over $X$ equipped with an inner product, and let $\tilde X \to X$ be the associated O(n)-principal bundle of orthonormal frames in $\xi$.

There is an equivalence in ∞Grpd

$Fun^\otimes(Bord_n^{(X,\xi)}, \mathcal{C}) \simeq Top_{O(n)}(\tilde X, \tilde \mathcal{C}) \,,$

where on the right we regard $\tilde C$ as a topological space carrying the canonical $O(n)$-action discussed above.

This is (Lurie, theorem. 2.4.18). The following is some aspects of the idea of the proof in (Lurie, p. 57).

###### Remark

In the language of ∞-actions (as discussed there), the space $Top_{O(n)}(\tilde X, \tilde \mathcal{C})$ is that of horizontal maps fitting into

$\array{ X && \longrightarrow && \tilde X//O(n) \\ & \searrow && \swarrow \\ && B SO(n) }$

where the left map is the classifying map for $\xi$ and the right one is the canonical one out of the homotopy quotient.

###### Idea of Proof of theorem 2.

Notice that for each point $x \colon \ast \to X$ there is an induced inclusion

$Bord_n^{fr} \stackrel{x}{\longrightarrow} Bord_n^{(X,\xi)}$

of the framed cobordisms, def. 1, into those of $(X,\xi)$-structure, def. 3, including those cobordisms whose map to $X$ is constant on $X$, and observing that for these an $(X,\xi)$-structure is equivalently an $n$-framing. Moreover, by corollary 1 the induced point evaluation is $O(n)$-equivariant, hence yielding a morphism of ∞-groupoids

$\alpha \;\colon\; Func^\otimes(Bord_n^{(X,\xi)}, \mathcal{C}) \longrightarrow Maps_{O(n)}(\tilde X, \tilde \mathcal{C}) \,,$

where $\tilde X$ denotes the $O(n)$-principal bundle to which $\xi$ is associated.

More generally, this is true for the pullback structure of $\xi$ along along any map $Y \to X$, yielding

$\alpha_Y \;\colon\; Func^\otimes(Bord_n^{(Y,\xi|Y)}, \mathcal{C}) \longrightarrow Maps_{O(n)}(\tilde X\underset{X}{\times} Y, \tilde \mathcal{C}) \,.$

By the previous comment, observe that $\alpha_Y$ is an equivalence for $Y = \ast$.

Now the codomain of this natural transformation sends (∞,1)-colimits in $Y$ over $X$ to (∞,1)-limits. (Lurie, theorem 3.1.8) shows that the same is true for the domain. Hence $\alpha_Y$ is an equivalence for all $Y$ that appear as (∞,1)-colimits of the point. But this is the case for all ∞-groupoids $Y$, by this proposition.

We consider now some special cases of the general definition of local structure-topological field theory

#### For framed cobordisms in a topological space

We discuss the special case of the cobordism hypothesis for $(X,\xi)$-cobordisms (def. 3) for the case that the vector bundle $\xi$ is the trivial vector bundle $\xi = \mathbb{R}^n \otimes X$.

In this case $\tilde X = O(n) \times X$. Write

$Bord_n^{fr}(X) := Bord_n^{(X,X \times \mathbb{R}^n)} \,.$

Write $\Pi(X) \in$ ∞Grpd for the fundamental ∞-groupoid of $X$.

###### Corollary

There is an equivalence in ∞Grpd

$Fun^\otimes(Bord^{fr}_n(X), C) \simeq (\infty,n)Cat(\Pi(X), \tilde C) \simeq \infty Grpd(\Pi(X), Core(\tilde C)) \,,$

This is a special case of the above theorem.

Notice that one can read this as saying that $Cob_n(X)$ is roughly like the free symmetric monoidal (∞,n)-category on the fundamental ∞-groupoid of $X$ (relative to $\infty$-categories of fully dualizable objects at least).

#### For cobordisms with $G$-structure

We discuss the special case of the cobordism hypothesis for $(X,\xi)$-bundles (def. 3) for the special case of G-structure (example 4), hence for the case that $X$ is the classifying space of a topological group.

Let $G$ be a topological group equipped with a homomorphism $\chi : G \to O(n)$ to the orthogonal group. Notice that via the canonical linear representation $\mathbf{B}O(n) \to$ Vect of $O(n)$ on $\mathbb{R}^n$, this induces accordingly a representation of $G$ on $\mathbb{R}^n$..

Let then

• $X := B G$ be the classifying space for $G$;

• $\xi_\chi := \mathbb{R}^n \times_G E G$ be the corresponding associated vector bundle to the universal principal bundle $E G \to B G$.

###### Definition

We say

$Bord^G_n := Bord_n^{(B G, \xi_\chi)} \,.$

is the $(\infty,n)$-category of cobordisms with $G$-structure.

See (Lurie, notation 2.4.21)

###### Definition

We have

• For $G = 1$ the trivial group, a $G$-structure is just a framing and so

$Bord_n^{(1,\xi)} \simeq Bord_n^{fr}$

reproduces the $(\infty,n)$-category of framed cobordisms, def. 1.

• For $G = SO(n)$ the special orthogonal group equipped with the canonical embedding $\chi : SO(n) \to O(n)$ a $G$-structure is an orientation

$Bord_n^{(SO(n))} \simeq Bord_n^{or} \,.$
• For $G = O(n)$ the orthogonal group itself equipped with the identity map $\chi : O(n) \to O(n)$ a $G$-structure is no structure at all,

$Bord_n^{O(n)} \simeq Bord_n \,.$

See (Lurie, example 2.4.22).

Then we have the following version of the cobordism hypothesis for manifolds with $G$-structure.

###### Corollary

For $G$ an ∞-group equipped with a homomorphism $G \to O(n)$ to the orthogonal group (regarded as an ∞-group in ∞Grpd), then evaluation on the point induces an equivalence

$Fun^\otimes( Bord_n^{G}, \mathcal{C} ) \simeq (\tilde {\mathcal{C}})^{G}$

between extended TQFTs on $n$-dimensional manifolds with G-structure and the ∞-groupoid homotopy invariants of the infinity-action of $G$ on $\tilde \mathcal{C}$ (which is induced by the evaluation on the point).

This is (Lurie, theorem 2.4.26).

###### Proof

Theorem 2 asserts that

$Fun^\otimes( Bord_n^{G}, \mathcal{C} ) \simeq Maps_{G}(E G , \tilde C) \,.$

Hence it remains to see that the right hand side are equivalently the homotopy invariants of the $G$-∞-action. This follows for instance with the discussion at ∞-action, by which

$Maps_G(V,W)\simeq \infty Grpd_{/B G}(V/\!/G, W/\!/G) \,.$

This yields

$Maps_{G}(E G , \tilde C) \simeq \infty Grpd_{/ B G}( B G, \tilde C /\!/B G ) \,.$

By the discussion at dependent product

$\infty Grpd_{/ B G}( B G, \tilde C /\!/B G ) \simeq \underset{B G}{\prod} (\tilde C /\!/B G)$

which are the homotopy invariants.

#### For (un-)oriented cobordisms

The case that $\chi \colon G \longrightarrow O(n)$ is the identity is at the other extreme of the framed case, and turns out to be similarly fundamental.

For $\mathbf{H}$ an (∞,1)-topos, write $Corr_n(\mathbf{H})^\otimes$ for the (∞,n)-category of correspondences in $\mathbf{H}$. For $Phases \in DCat_n(\mathbf{H})$ an (∞,n)-category with duals internal to $\mathbf{H}$, write $Corr_n(\mathbf{H}_/{Phases})^{\otimes_{phases}}$ for the (∞,n)-category of correspondences over $Phases$ and equipped with the phased tensor product. There is the forgetful monoidal (∞,n)-functor

$Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \longrightarrow Corr_n(\mathbf{H})^\otimes$

By the discussion at (∞,n)-category of correspondences these are (∞,n)-categories with duals and the canonical $O(n)$-∞-action on them, corollary 1, is trivial for $Corr_n(\mathbf{H})$. This means that an $O(n)$-homotopy fixed point in $Corr_n(\mathbf{H})$ is just an object of $\mathbf{H}$ equipped in turn with an $O(n)$-∞-action. Therefore

###### Proposition

Local unoriented-topological field theory

$Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H})^\otimes$

are equivalent to objects $X \in \mathbf{H}$ equipped with an $O(n)$-∞-action.

At least for $\mathbf{H} =$ ∞Grpd, then given such, the corresponding field theory $Z_{X/\!/O(n)}$ sends a cobordism $\Sigma$ to the space of maps

$\array{ \Pi(\Sigma) && \longrightarrow && X//O(n) \\ & {}_{\mathllap{T \Sigma \oplus \mathbb{R}^{n-dim(\Sigma)}}}\searrow && \swarrow \\ && B O(n) }$

hence

$Z_{X//O(n)} \colon \Sigma \mapsto [\Pi(\Sigma),X]^{O(n)} \,.$

In particular this means that the assignment to the point is again $X$ itself.

This is a slight rephrasing of the paragraph pp 58-59 in (Lurie).

###### Proposition

At least for $\mathbf{H} =$ ∞Grpd, with $X \in \mathbf{H}$ an object equipped with an $O(n)$-∞-action, then horizontal lifts in

$\array{ && Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \\ & \nearrow & \downarrow \\ Bord_n^\sqcup &\underset{X//O(n)}{\longrightarrow}& Corr_n(\mathbf{H})^\otimes }$

are equivalent to

$(Bord_n^{(X//O(n), X \underset{O(n)}{\times}\mathbb{R}^n)})^{\sqcup} \longrightarrow Phases^\otimes \,.$

This is (Lurie, prop. 3.2.8).

###### Remark

Via the interpretation of local field theories with coefficients in $Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$ as Local prequantum field theory, the statement of prop. 4 translates in quantum field theory jargon to the statement that “All background structures are fields.” This is essentially the slogan of general covariance.

###### Corollary

Let $Phases^\otimes \in Ab_\infty(\mathbf{H})$ be an abelian ∞-group object, regarded as a (∞,n)-category with duals internal to $\mathbf{H}$.

At least if $\mathbf{H} =$ ∞Grpd, then local unoriented-topological field theories of the form

$Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$

are equivalent to a choice

1. of $X \in \mathbf{H}$ equipped with an $O(n)$-∞-action

2. a homomorphism of $O(n)$-∞-actions $L \colon X \to Phases$ (where $Phases^\otimes$ is equipped with the canonical $\infty$-action induced from the framed cobordism hypothesis), hence (by the discussion at ∞-action) to a horizontal morphism in $\mathbf{H}$ fitting into the diagram

$\array{ X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n) \\ & \searrow &\swArrow_\simeq& \swarrow \\ && B O(n) } \,.$
###### Proof

By prop. 3 the co-restriction

$Bord_n^\sqcup \stackrel{Z}{\longrightarrow} Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \longrightarrow Corr_n(\mathbf{H})^\otimes$

is equivalent to an ∞-action

$\array{ X &\longrightarrow& X//O(n) \\ && \downarrow \\ && B O(n) }$

Therefore by prop. 4 $Z$ is equivalent to

$(Bord_n^{(X//O(n), X \underset{O(n)}{\times} \mathbb{R}^n) })^\sqcup \longrightarrow Phases^\otimes \,.$

Finally, by theorem 2 and in view of remark 1, this is equivalent to maps of the form

$\array{ X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n) \\ & \searrow && \swarrow \\ && B O(n) } \,.$

By the discussion at Local prequantum field theory, these statements hold also for fields with moduli spaces in more general $(\infty,1)$-toposes $\mathbf{H}$ (one sufficient condition is that $\mathbf{H}$ has an (infinity,1)-site of definition all whose objects are etale contractible).

Some examples are discussed at prequantum field theory in the section Higher Chern-Simons field theory – Levels.

#### For HQFTs

If in def. 3 one chooses $X = B SO(n) \times Y$ for any topological space $Y$, and $\xi$ the pullback of the canonical vector bundle bundle on $B SO$ to $B SO \times Y$, then an $(\infty,n)$-functor $Bord^{X}_n \to C$ is similar to what Turaev calls an HQFT over $Y$.

### For cobordisms with singularities (boundaries/branes and defects/domain walls)

There is a vast generalization of the plain $(\infty,n)$-category of cobordisms (with topological structure) considered above given by allowing the cobordisms to be equipped with various types of singularities (Lurie 09, Definition Sketch 4.3.2).

Each type of singularity in dimension $k$ now corresponds to a new generator k-morphisms, and the (framed) $(\infty,n)$-category of cobordisms with singularities is now no longer the free symmetric monoidal $(\infty,n)$-category freely generated from just a point (a 0-morphisms), but freely generated from these chosen generators. This general version is (Lurie 09, Theorem 4.3.11).

For instance if the generator on top of the point $\ast$ is a 1-morphism of the form $\emptyset \to \ast$, then this defines a type of codimension $(n-1)$-boundary; and hence extended TQFTs with such boundary data and with coefficients in some symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ with all dual are equivalent to choices of morphisms $1 \to A$, where $A \in \mathcal{C}$ is the fully dualizable object assigned to the point, as before, and now equipped with a morphism from the tensor unit into it. Indeed, this is the usual datum that describes branes in QFT (see for instance at FRS formalism).

For more on this see at QFT with defects.

### For noncompact cobordisms

One important variant of the category of cobordisms is obtained by discarding all those morphisms which have non-empty incoming (say, dually one could use outgoing) bounrary component. Then a representation of this category imposes on its values “cups but no caps”, hence only half of the data of a dualizable object in the given degree.

Accordingly, in this case the cobordism hypothesis says that such a functor is given not quite by a fully dualizable object, but by a weaker structure called a Calabi-Yau object (see there for more).

2-dimensional TQFT of this form is known as TCFT, see there for more

### For cobordisms with geometric structure

A non-topological quantum field theory is a representation of a cobordism category for cobordisms equipped with extra stuff, structure, property that is “not just topological”, meaning roughly not of the form of def. 3.

The theory for this more general case is not as far developed yet.

• steps towards classification of quantum field theories with super-Euclidean structure are discussed at

• a general definition of a cobordism category of cobordisms equipped with geometric structure given by a morphism into, roughly, a smooth infinity-groupoid of structure is discussed in (Ayala).

## Remarks

### Morphisms of TQFTs

In particular this means that $Fun^\otimes(Bord_n^{fr} , C )$ is itself an $(\infinity,0)$-category, i.e. an ∞-groupoid.

When interpreting symmetric monoidal functors from bordisms to $C$ as TQFTs this means that TQFTs with given codomain $C$ form a space, an ∞-groupoid. In particular, any two of them are either equivalent or have no morphism between them.

According to Chris Schommer-Pries interesting morphisms of TQFTs arise when looking at transformations only on sub-categories on all of $Bord_n$. This is described at QFT with defects .

### Invariants determined from the point

The theorem does say that the invariant attached by an extended TQFT to the point determines all the higher invariants – however it is important to notice that there are strong constraints on what is assigned to the point. For an $n$-dimensional framed theory one needs to assign a fully dualizable object, and the meaning of the term “fully dualizable” depends on $n$, and gets increasingly hard to satisfy as n grows..

For an $n$-dimensional unoriented theory, the object assigned to the point has to be a fixed point for the $O(n)$- action on fully dualizable objects that is obtained from the framed case of the theorem.

In the 1d case, this $O(1)$ action on dualizable objects takes every object to its dual, and an $O(1)$ fixed point is indeed a vector space with a nondegenerate symmetric inner product.

For an oriented theory $n$-dimensional theory need an $SO(n)$-fixed point, which for $n=1$ is nothing but for $n=2$ ends up meaning a Calabi-Yau category (in the case the target 2-category is that of categories).

In fact something more general is true: if one wants a theory that takes values on manifolds equipped with a $G$-structure, for $G$ any group mapping to $O(n)$ (such as for instance orientation already discussed or its higher versions Spin structure or String structure or Fivebrane structure or …) one needs to assign to the point a $G$-fixed point in dualizable objects in your category (with $G$ acting through $O(n)$).

This beautifully includes all the above plus for example manifolds with maps (up to homotopy) to some auxiliary (connected) space $X$ – here we take $G$ to be the loop space $\Omega X$ of $X$ (mapping trivially to $O(n)$), so that a reduction of the structure group of the manifold to $G$ involves a map to the delooping $\mathcal{B}G \simeq X$.

Such theories are classified by $X$-families of fully dualizable objects.

Notice that there is an important subtlety of Lurie’s theorem in the case of manifolds with $G$-structure which is easy to confuse. The general version of the theorem about TFTs does not say that they are the $G$-fixed points for the $G$-action on fully dualizable objects, but rather they are the homotopy fixed points. This is very important because a homotopy fixed point is not just a property. It is additional structure. Depending on $G$, this additional structure is often encoded in the higher dimensional portion of the field theory.

One can see this in the 1 dimensional case: there is no property of vector spaces which automatically endows them with an inner product, but it is extra structure.

duality between algebra and geometry in physics:

## References

The original hypothesis is formulated in

The formalization and proof is described in

This is almost complete, except for one step that is not discussed in detail. But a new (unpublished) result by Søren Galatius bridges that step in particular and drastically simplifies the whole proof in general.

The comparatively simple case of $n = 1$ is spelled out in detail in

and aspects of the case $n = 2$ (see also at 2d TQFT) are discussed in

Lecture notes and reviews on the topic of the cobordisms hypothesis include

Discussion of the canonical $O(n)$-action on n-fold loop spaces (which may be thought of as a special case of the cobordism hypothesis) includes

• Gerald Gaudens, Luc Menichi, section 5 of Batalin-Vilkovisky algebras and the $J$-homomorphism, Topology and its Applications Volume 156, Issue 2, 1 December 2008, Pages 365–374 (arXiv:0707.3103)

Cobordisms with geometric structure are discussed in

Revised on May 7, 2015 18:44:15 by Urs Schreiber (78.102.213.68)