03.16.03 · modern-geometry / tqft

Extended TQFT and the cobordism hypothesis

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Lurie 2009 On the Classification of Topological Field Theories (Current Developments in Mathematics); Baez–Dolan 1995 J. Math. Phys. 36; Freed CBMS Lectures on Field Theory and Topology Lecture 5; Calaque–Scheimbauer 'A note on the (∞,n)-category of cobordisms'

Intuition Beginner

A topological field theory is a rule that turns shapes into algebra. Feed it the empty space and a circle, and it answers with numbers and vector spaces; feed it a tube connecting two circles, and it answers with a linear map. The Atiyah–Segal picture stops at one level down from a top-dimensional shape: circles and surfaces, say, for a two-dimensional theory.

Extending the theory means refusing to stop there. You keep cutting. A surface cuts into tubes; a tube cuts into shorter tubes; the circles bounding them cut into arcs; and the arcs end at points. The idea is to ask the field theory to assign meaning to every one of these pieces, all the way down to single points, in a way that is consistent every time you reglue.

Why bother? Because if you can cut spacetime all the way down to points, then the whole field theory is locked in by what it does to one point. Locality, taken to its logical extreme, says one richly structured algebraic object at a point determines everything above it.

There is a payoff for this stubbornness. Once the theory is fully local, building it becomes a one-step job. You no longer check a long list of gluing rules; you supply a single object with enough internal symmetry, and the rest is forced. The cobordism hypothesis is the precise promise that this works.

Visual Beginner

Alt text: A vertical ladder of geometric pieces. At the top sits a two-dimensional surface with boundary. An arrow down shows it sliced into cylindrical tubes. A further arrow slices a tube into a circle, then the circle into two arcs, and finally the arcs terminate at a single dot at the bottom. A bracket on the right spans the whole ladder and points to the bottom dot, captioned "value here decides everything above." The picture conveys that an extended field theory assigns data at every level of cutting, and that the data on a point propagates upward to fix the entire theory.

Worked example Beginner

Take the simplest case: a one-dimensional theory, where the top shapes are intervals and circles, and one level down sits the point.

An ordinary one-dimensional field theory already assigns a vector space $V$ to the point and reads a circle as a single number, the dimension of $V$ . The extended version asks for more: to the point it assigns the vector space $V$ itself, to an interval it assigns the pairing that turns $V$ and its dual into a number, and to the reversed interval it assigns the copairing that produces an element of $V$ tensored with its dual.

For this to close up consistently, $V$ must be finite-dimensional. Then the two arcs of a circle glue back to give exactly the dimension of $V$ , matching the unextended answer. Try it with $V$ four-dimensional: the point carries a four-dimensional space, the pairing and copairing are the obvious ones, and regluing the circle returns the number $4$ .

What this tells us: in dimension one, a fully local field theory is the same thing as a finite-dimensional vector space, supplied once at the point. The finiteness is not a side condition you impose; it is what makes the arcs reglue at all.

Check your understanding Beginner

Formal definition Intermediate+

Fix a base symmetric monoidal $(\infty, n)$ -category $C$ ; informally, an $(\infty, n)$ -category has objects, $1$ -morphisms, up to $n$ -morphisms, and above level $n$ only invertible morphisms (homotopies). A rigorous model is a complete $n$ -fold Segal space ^{[Lurie §2]}; the reader unfamiliar with that model may treat $C$ as "a category with morphisms between morphisms, $n$ levels deep, and everything reversible thereafter." The unextended target $Vect_{C}$ of 03.16.01 is the case $n = 1$ .

The extended bordism category. For each $0 \leq k \leq n$ , the framed bordism $(\infty, n)$ -category $Bord_{n}^{f r}$ has as objects framed $0$ -manifolds (finite sets of framed points), as $1$ -morphisms framed $1$ -manifolds with boundary, and in general as $k$ -morphisms framed $k$ -manifolds with corners, for $k \leq n$ . The $n$ -morphisms are top bordisms; above level $n$ the morphisms are framed diffeomorphisms, then isotopies between diffeomorphisms, and so on — all invertible. A framing is a chosen identification of the (stabilised) tangent bundle with the constant bundle; replacing it by a reduction of structure group along $H \to O (n)$ gives the $H$ -structured variant $Bord_{n}^{H}$ , with $H = S O (n)$ orientations and $H = Spin (n)$ spin structures as in 03.16.01. Disjoint union makes $Bord_{n}^{f r}$ symmetric monoidal with unit the empty manifold.

Definition (fully extended TQFT). A fully extended $n$ -dimensional framed topological field theory valued in $C$ is a symmetric monoidal functor $$ Z : \mathrm{Bord}_n^{fr} \longrightarrow \mathcal C . $$ Restricting attention to the top two levels recovers an Atiyah–Segal theory: $Z$ on closed $(n - 1)$ -manifolds gives state spaces and on closed $n$ -manifolds gives partition numbers. The extension is the data of $Z$ on all lower-dimensional manifolds and corners.

Dualizability. An object $X$ of a symmetric monoidal $(\infty, n)$ -category $C$ is dualizable if there is a dual $X^{\lor}$ with evaluation $ev : X \otimes X^{\lor} \to 1$ and coevaluation $coev : 1 \to X^{\lor} \otimes X$ satisfying the zig-zag (triangle) identities up to coherent higher morphisms. $X$ is fully dualizable if it is dualizable and, in addition, the morphisms $ev$ and $coev$ themselves admit both left and right adjoints, those adjoints admit adjoints, and so on, until level $n$ is reached. Write $C^{f d}$ for the full subcategory of fully dualizable objects, and $(C^{f d})^{\sim}$ for its maximal $\infty$ -groupoid: discard all non-invertible morphisms, keeping objects, equivalences, and the homotopies among them.

The point object of $Bord_{n}^{f r}$ — a single positively framed point, denoted $pt_{+}$ — is itself fully dualizable, its dual being the oppositely framed point, with the evaluation and coevaluation realised by the interval read in two directions and their adjoints supplied by higher bordisms. Any symmetric monoidal functor $Z$ sends fully dualizable objects to fully dualizable objects, so $Z (pt_{+}) \in C^{f d}$ .

Counterexamples to common slips

Dualizable is weaker than fully dualizable. In $Vect_{C}$ (the $n = 1$ case) every finite-dimensional space is dualizable, and there is nothing higher to check, so the two notions coincide. At $n = 2$ they part ways: an algebra is dualizable as soon as it is finite-dimensional, but full dualizability also requires the multiplication and its adjoint to be well-behaved — the separable / Calabi–Yau condition. Conflating the two over-counts the theories.
Framed is not oriented. A framed theory needs no compatibility with rotations of the tangent space; an oriented theory does. The point object of $Bord_{n}^{S O}$ is an $S O (n)$ -homotopy-fixed point of $C^{f d}$ , a strictly stronger datum than a bare fully dualizable object. Forgetting this confuses the framed classification with the oriented one.
" $(\infty, n)$ " is not " $n$ -dimensional vectors." The $\infty$ records homotopies between morphisms at every level, not a dimension count. A complete $1$ -fold Segal space is an ordinary $\infty$ -category, already infinite-dimensional in this sense even for $n = 1$ .

Key theorem with proof Intermediate+

Theorem (cobordism hypothesis, framed version; Baez–Dolan conjecture, Lurie). Let $C$ be a symmetric monoidal $(\infty, n)$ -category. Evaluation at the positively framed point, $$ \mathrm{ev}_{\mathrm{pt}} : \mathrm{Fun}^{\otimes}!\big(\mathrm{Bord}n^{fr},, \mathcal C\big) \longrightarrow \big(\mathcal C^{fd}\big)^{\sim}, \qquad Z \longmapsto Z(\mathrm{pt}+), $$ is an equivalence of $\infty$ -groupoids. In words: a fully extended framed $n$ -dimensional TQFT valued in $C$ is freely determined by a single fully dualizable object of $C$ , and every fully dualizable object arises ^{[Lurie §2.4]}.

Proof (structure). The content is concentrated in three claims, each of which Lurie establishes by an induction on dimension; the present account states them at the level of structure, which is the appropriate altitude for a frontier survey.

Faithfulness — a theory is determined by its point. Given $Z$ , the framed point $pt_{+}$ and its dual generate $Bord_{n}^{f r}$ under the symmetric monoidal operations, taking adjoints, and gluing. Every framed bordism can be assembled from finitely many copies of $pt_{+}$ by these moves — a Morse-theoretic / handle decomposition statement for manifolds with corners. Since $Z$ is symmetric monoidal and preserves the relevant adjunctions (a formal consequence of functoriality on a higher category), its value on any bordism is computed from $Z (pt_{+})$ . Hence $Z$ is recovered from $Z (pt_{+})$ , and two theories agreeing at the point agree everywhere: $ev_{pt}$ is faithful.

Well-definedness of the target — the point lands in $C^{f d}$ . Because $pt_{+}$ is fully dualizable in $Bord_{n}^{f r}$ and symmetric monoidal functors preserve full dualizability, $Z (pt_{+})$ is fully dualizable. Thus $ev_{pt}$ does map into $(C^{f d})^{\sim}$ .

Surjectivity — every fully dualizable object integrates to a theory. This is the substantive half. One must show that given a fully dualizable $X \in C$ , the assignment $pt_{+} \mapsto X$ extends, uniquely up to coherent homotopy, to a symmetric monoidal functor on all of $Bord_{n}^{f r}$ . Lurie's argument is an induction that builds the functor on $k$ -morphisms from its values on $(k - 1)$ -morphisms, the existence of the adjoints packaged in full dualizability being exactly what permits each inductive step; the framed bordism category is, in a precise sense, the free symmetric monoidal $(\infty, n)$ -category on one fully dualizable object, and the theorem is the corresponding universal property ^{[Lurie §3]}. The induction is controlled by an analysis of the spaces of framed bordisms via the cobordism category of Galatius–Madsen–Tillmann–Weiss, which identifies $Bord_{n}^{f r}$ with a highly connected space whose homotopy type carries the framing data. $■$

Bridge. This theorem builds toward the entire stable-homotopy classification of field theories and is dual to the gluing picture of 03.16.01: where Atiyah–Segal computes a theory by cutting along closed hypersurfaces, the cobordism hypothesis computes it by cutting all the way to points, and the foundational reason the two agree is that the point generates everything above it. This is exactly the statement that locality, pushed to its limit, trades a functor for a single object. The framed case generalises to structured theories through the $O (n)$ -action introduced next, and the invertible sub-case appears again in 03.16.04, where $C^{f d}$ is replaced by its Picard groupoid and the classifying object becomes a spectrum. Putting these together, the hypothesis is the bridge by which the algebra of one object and the topology of all bordisms become the same data.

Exercises Intermediate+

Exercise (hard, short-answer). The framing of $\mathrm{pt}_+$ can be rotated by an element of $O(n)$. Explain how this produces an $O(n)$-action on $(\mathcal C^{fd})^\sim$ and why oriented theories are its $SO(n)$-fixed points rather than arbitrary points.

Hint

A change of framing is an automorphism of $pt_{+}$ in the bordism category; functoriality transports it to an automorphism of the image object.

Answer

Rubric. Changing the framing of $pt_{+}$ by $g \in O (n)$ is a self-equivalence of $pt_{+}$ in $Bord_{n}^{f r}$ ; under any $Z$ it becomes a self-equivalence of $Z (pt_{+})$ , and these assemble into an action of the topological group $O (n)$ on the space $(C^{f d})^{\sim}$ . An oriented theory is one whose point data is compatibly invariant under the rotation subgroup $S O (n)$ — a homotopy-fixed point, carrying the coherence data of the invariance, not a mere fixed object. Full credit derives the action from framing automorphisms and distinguishes a homotopy-fixed point from a strict fixed point.

Exercise (hard, short-answer). Sketch why surjectivity (every fully dualizable object integrates to a theory) is the hard half of the cobordism hypothesis, while faithfulness is comparatively formal.

Hint

Faithfulness uses only that the point generates; surjectivity must build a coherent functor on infinitely many corners.

Answer

Rubric. Faithfulness follows once $pt_{+}$ generates $Bord_{n}^{f r}$ under monoidal operations, gluing, and adjoints: the value of $Z$ on anything is then a formula in $Z (pt_{+})$ . Surjectivity demands the reverse — extending a chosen object to a fully coherent symmetric monoidal functor on all $k$ -morphisms for $k \leq n$ , with every adjunction and higher coherence supplied — proved by a dimension induction whose each step consumes one layer of the full-dualizability data. Full credit contrasts "the point generates" (formal) with "the assignment extends coherently" (the induction).

Advanced results Master

The tangential-structure theorem. Let $H \to O (n)$ be a map of topological groups, defining an $H$ -structure on $n$ -manifolds (orientation for $S O (n)$ , spin for $Spin (n)$ , framing for the identity-only group). The bordism category $Bord_{n}^{H}$ classifies $H$ -structured fully extended theories, and the cobordism hypothesis acquires its general form: such theories are classified by the homotopy-fixed points $((C^{f d})^{\sim})^{h H}$ of the $H$ -action obtained by restricting the $O (n)$ -action along $H \to O (n)$ ^{[Lurie §2.4]}. The framed case is $H$ the identity-only group, where the fixed-point space is the whole of $(C^{f d})^{\sim}$ . Orientations and spin theories are the two cases of immediate physical interest, and the homotopy-fixed-point data is precisely the extra coherence (an $S O (n)$ - or $Spin (n)$ -equivariant cancellation of the framing anomaly) that an unframed theory must carry.

Low-dimensional verifications. For $n = 1$ the statement reduces to the identification of dualizable objects, classical and complete. For $n = 2$ the framed and oriented cases were established directly by Schommer-Pries and by Hopkins–Lurie: the fully dualizable objects of the relevant $2$ -category of algebras are the finite-dimensional separable algebras, and the oriented $S O (2)$ -fixed-point condition selects exactly the commutative Frobenius algebras of 03.16.02, with the $S O (2) ≃ S^{1}$ rotation acting through the Serre automorphism. The $n = 2$ case is therefore both a verification of the hypothesis and a re-derivation of the Frobenius-algebra theorem from a structural principle rather than a generators-and-relations presentation.

Invertible theories and spectra. When one restricts to the invertible part — theories whose point object is $\otimes$ -invertible and whose every bordism value is an equivalence — the classifying groupoid collapses to the Picard $\infty$ -groupoid $Pic (C)$ , which is an infinite loop space and so the zeroth space of a connective spectrum. The cobordism hypothesis for invertible theories becomes a statement in stable homotopy theory: invertible $H$ -theories are classified by maps of spectra out of a Thom spectrum built from $H$ , a translation that turns the classification of anomalies into a computation in generalised cohomology. This is the entry point to 03.16.04 and the reason the spectrum object of 03.12.04 is the natural home for the invertible case.

Synthesis. The cobordism hypothesis is the central insight that organises the whole extended theory: it is dual to the Atiyah–Segal gluing picture of 03.16.01, it generalises the $2$ d Frobenius classification of 03.16.02 to all dimensions and all tangential structures, and it is exactly the universal property of $Bord_{n}^{H}$ as a free symmetric monoidal $(\infty, n)$ -category. Putting these together, three apparently different problems — building a local field theory, classifying its anomalies, computing its partition function — collapse onto one object at a point and the symmetries that act on it. The foundational reason a framed theory is rigid is that the framed point generates the bordism category freely, so a functor out of it is no more and no less than the image of that generator; the tangential-structure refinement then records, as homotopy-fixed-point data, precisely the rotation anomaly that an unframed theory must cancel, and the invertible specialisation is dual to a computation in the stable homotopy of a Thom spectrum, which is where 03.16.04 continues the story.

Full proof set Master

Proposition (the point object is fully dualizable in $Bord_{n}^{f r}$ ). The positively framed point $pt_{+}$ is a fully dualizable object of $Bord_{n}^{f r}$ , with dual the negatively framed point $pt_{-}$ .

Proof. The evaluation $ev : pt_{+} ⊔ pt_{-} \to \emptyset$ is the right-half interval (a framed $1$ -bordism from two points to the empty $0$ -manifold), and the coevaluation $coev : \emptyset \to pt_{-} ⊔ pt_{+}$ is the left-half interval. The two zig-zag composites are framed intervals isotopic, rel endpoints, to the identity bordism on $pt_{+}$ and on $pt_{-}$ ; the isotopies are the higher morphisms witnessing the triangle identities. This establishes dualizability. For full dualizability one must produce left and right adjoints to $ev$ and $coev$ , then adjoints to those, up to level $n$ . Each required adjoint is realised by a bordism with corners obtained from the interval by attaching elementary handles, the adjunction units and counits being saddle bordisms; the existence of the whole tower up to level $n$ is the statement that every framed manifold with corners of dimension $\leq n$ admits a handle decomposition compatible with the framing. Hence $pt_{+}$ is fully dualizable. $■$

Proposition (functors preserve full dualizability). If $F : C \to D$ is symmetric monoidal and $X \in C$ is fully dualizable, then $F (X)$ is fully dualizable in $D$ .

Proof. A symmetric monoidal functor preserves duals: $F (X^{\lor})$ is a dual of $F (X)$ with $F (ev)$ and $F (coev)$ as the structure maps, the zig-zag identities being images of those for $X$ . Full dualizability is the existence of an adjoint tower above $ev$ and $coev$ ; a functor between $(\infty, n)$ -categories carries an adjunction $(f ⊣ g)$ to an adjunction $(F f ⊣ F g)$ , because adjunctions are characterised by unit and counit morphisms satisfying the triangle identities, all of which $F$ preserves. Applying this at each level of the tower for $X$ produces the tower for $F (X)$ . $■$

Proposition ( $Z (S^{1}) = dim Z (pt_{+})$ in the $1$ -categorical truncation). For a fully extended theory $Z$ valued in a symmetric monoidal $(\infty, 1)$ -category and $X = Z (pt_{+})$ , the value on the circle is the categorical dimension $tr (id_{X}) = ev_{X} \circ τ \circ coev_{X}$ .

Proof. The circle decomposes as a coevaluation $1$ -bordism $\emptyset \to pt_{-} ⊔ pt_{+}$ followed by the symmetry braiding the two points and then an evaluation $pt_{+} ⊔ pt_{-} \to \emptyset$ . Applying the symmetric monoidal functor $Z$ and using $Z (pt_{\pm}) = X, X^{\lor}$ together with the previous propositions sends this composite to $ev_{X} \circ τ_{X, X^{\lor}} \circ coev_{X}$ , which is the categorical trace of the identity, i.e. the dimension of $X$ . In $Vect_{C}$ this is $dim X$ , recovering $Z (S^{1}) = dim Z (pt_{+})$ and the gluing relation of 03.16.01. $■$

The full surjectivity half of the cobordism hypothesis — that every fully dualizable object integrates to a theory — and the tangential-structure homotopy-fixed-point refinement are stated above without proof; see Lurie ^{[Lurie §3]} for the inductive construction and Freed ^{[Freed 2013]} for the expository account of both.

Connections Master

The Atiyah–Segal axioms 03.16.01. That unit defines a TQFT as a symmetric monoidal functor on the $1$ -categorical bordism category, with state spaces, partition functions, and the gluing law. The present unit is its downward extension: the same functorial idea pushed to an $(\infty, n)$ -category whose objects are points. The gluing law there becomes the locality principle here, and $Z (S^{1}) = dim Z (Y)$ is the truncation of the categorical-trace computation proved above.
Classification of 2d oriented TQFTs 03.16.02. The Frobenius-algebra theorem is the $n = 2$ , oriented sanity check of the cobordism hypothesis: the fully dualizable algebras are the separable ones, and the $S O (2)$ -fixed-point condition cuts them down to the commutative Frobenius algebras classified there. The hypothesis re-derives that result from a universal property rather than a generators-and-relations presentation of surfaces.
Spectrum 03.12.04. Restricting to invertible theories replaces the target by its Picard $\infty$ -groupoid, an infinite loop space, hence the zeroth space of a connective spectrum. The classification of invertible extended theories thereby becomes a stable-homotopy computation, and the spectrum object built there is the natural codomain. This is the bridge into the Madsen–Tillmann / Anderson-dual classification.
Invertible field theories and the Freed–Hopkins classification 03.16.04. The invertible specialisation of the cobordism hypothesis is precisely the input to that unit: once invertible theories are maps of spectra out of a Thom spectrum, the deformation classes of $n$ -dimensional invertible $H$ -theories are computed as a generalised cohomology group, which is the Freed–Hopkins theorem.

Historical & philosophical context Master

The hypothesis was formulated by John Baez and James Dolan in "Higher-dimensional algebra and topological quantum field theory" (Journal of Mathematical Physics 36, 6073–6105, 1995), as one of a family of conjectures — the tangle hypothesis and the cobordism hypothesis — organising extended TQFT through $n$ -category theory; their stated form proposed that the framed bordism $n$ -category is the free stable $n$ -category with duals on one object ^{[Baez–Dolan 1995]}. A complete formulation and proof sketch was given by Jacob Lurie in "On the Classification of Topological Field Theories" (Current Developments in Mathematics 2008, published 2009), which recast the conjecture in the language of $(\infty, n)$ -categories modelled by complete $n$ -fold Segal spaces and reduced it to a universal property of the bordism category ^{[Lurie §1]}. The framing analysis rests on the cobordism-category theorem of Galatius, Madsen, Tillmann, and Weiss, which identifies the homotopy type of bordism categories with infinite loop spaces of Thom spectra.

Daniel Freed's "The cobordism hypothesis" (Bulletin of the American Mathematical Society 50, 57–92, 2013) supplied the expository synthesis aimed at the mathematical-physics community, and his CBMS lectures developed the consequences for invertible theories and anomalies ^{[Freed 2013]}. The two-dimensional case was verified independently by Christopher Schommer-Pries and, in the structured setting, by Michael Hopkins and Lurie, where the Serre automorphism of a Calabi–Yau algebra is identified with the generator of the $S O (2)$ -action.

Bibliography Master

@article{BaezDolan1995,
  author  = {Baez, John C. and Dolan, James},
  title   = {Higher-dimensional algebra and topological quantum field theory},
  journal = {Journal of Mathematical Physics},
  volume  = {36},
  number  = {11},
  pages   = {6073--6105},
  year    = {1995}
}

@incollection{Lurie2009Classification,
  author    = {Lurie, Jacob},
  title     = {On the Classification of Topological Field Theories},
  booktitle = {Current Developments in Mathematics 2008},
  publisher = {International Press},
  pages     = {129--280},
  year      = {2009}
}

@article{Freed2013Cobordism,
  author  = {Freed, Daniel S.},
  title   = {The cobordism hypothesis},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {50},
  number  = {1},
  pages   = {57--92},
  year    = {2013}
}

@book{Freed2019CBMS,
  author    = {Freed, Daniel S.},
  title     = {Lectures on Field Theory and Topology},
  series    = {CBMS Regional Conference Series in Mathematics},
  number    = {133},
  publisher = {American Mathematical Society},
  year      = {2019}
}

@article{GMTW2009,
  author  = {Galatius, S{\o}ren and Madsen, Ib and Tillmann, Ulrike and Weiss, Michael},
  title   = {The homotopy type of the cobordism category},
  journal = {Acta Mathematica},
  volume  = {202},
  number  = {2},
  pages   = {195--239},
  year    = {2009}
}

@article{SchommerPries2014,
  author  = {Schommer-Pries, Christopher J.},
  title   = {Dualizability in low-dimensional higher category theory},
  journal = {Contemporary Mathematics},
  volume  = {613},
  pages   = {111--176},
  year    = {2014}
}

@book{Lurie2017HA,
  author    = {Lurie, Jacob},
  title     = {Higher Algebra},
  publisher = {Available at the author's website},
  year      = {2017},
  note      = {Chapter 4, dualizable objects and the Picard groupoid}
}

Prerequisites

03.16.01 pending
03.16.02 pending
03.12.04 pending

Tier anchors

beginner: Baez 'The Tale of n-Categories' (week 73–100, informal); Freed CBMS Lectures on Field Theory and Topology, Lecture 5 opening
intermediate: Lurie — On the Classification of Topological Field Theories §1–2; Freed CBMS Lectures, Lecture 5; Schommer-Pries 'Dualizability in Low-Dimensional Higher Category Theory'
master: Lurie 2009 On the Classification of Topological Field Theories (Current Developments in Mathematics); Baez–Dolan 1995 J. Math. Phys. 36; Freed CBMS Lectures on Field Theory and Topology Lecture 5; Calaque–Scheimbauer 'A note on the (∞,n)-category of cobordisms'

References

03-modern-geometry/Freed-AspectsOfFieldTheory.pdf · Extended locality, the value on a point, dualizability (Lecture 5 analogue)
Lurie — On the Classification of Topological Field Theories (2009) · Current Developments in Mathematics 2008, §1–3 (the cobordism hypothesis)
Baez and Dolan — Higher-dimensional algebra and topological quantum field theory (1995) · J. Math. Phys. 36, 6073–6105 (the tangle / cobordism hypotheses)
Freed — The cobordism hypothesis (2013) · Bull. Amer. Math. Soc. 50, 57–92

Estimated time

beginner: 16m
intermediate: 40m
master: 82m