Trivial Conormal Bundle in Geometry & Topology

• Trivial conormal bundle is a geometric condition where the conormal bundle admits a global nonzero section, making it isomorphic to a product bundle.
• It plays a key role in classifying immersions and embeddings, particularly by linking normal bundle triviality to self-intersection invariants and cobordism obstructions.
• In Legendrian and contact topology, trivial conormal bundles enable explicit algebraic models and simplify computations of invariants such as string topology and Legendrian contact homology.

A trivial conormal bundle refers to the geometric and topological condition in which the conormal bundle of a submanifold or the normal bundle of an immersion or embedding admits a global, nowhere-vanishing section, thus being isomorphic to a product bundle. In the context of smooth immersions $\beta : M^n \rightarrow Z^{n+1}$ between compact manifolds, the normal line bundle $\nu^\beta = \beta^* T Z / T M$ is trivial if $\nu^\beta \cong M \times \mathbb{R}$. Equivalently, the associated conormal bundle $\mu^\beta$, being the annihilator of $T M$ in $\beta^* T^* Z$, is trivial if and only if $\nu^\beta$ is trivial, since the dual of a real line bundle is trivial if and only if the original bundle is trivial. Triviality of the (co)normal bundle is a crucial property in several geometric, topological, and contact-topological constructions, appearing centrally in the paper of cobordisms, string topology, and invariants of Legendrian submanifolds.

1. Definitions and Fundamental Properties

For a compact or closed submanifold $K$ of a Riemannian manifold $Q$, the conormal bundle $N^* K$ is defined by

$$N^* K = \{ (q,p) \in T^* Q \mid q \in K,\; p|_{T_q K} = 0 \}.$$

When equipped with a Riemannian metric, the duality between the normal bundle $T K^\perp$ and the conormal bundle yields $N^* K \cong K \times \mathbb{R}^d$ if and only if the normal bundle $T K^\perp \cong K \times \mathbb{R}^d$ is trivial, for codimension $d$. The unit conormal bundle $\Lambda_K = N^* K \cap S^* Q$ then becomes diffeomorphic to $K \times S^{d-1}$ in the trivial case. For immersions or embeddings $\beta : M \rightarrow Z$ of codimension 1, the normal line bundle $\nu^\beta$ is trivial if and only if the conormal line bundle $\mu^\beta = (\nu^\beta)^*$ is trivial, i.e., $\mu^\beta \cong M \times \mathbb{R}$ (Katz, 2023).

2. Classification of Immersions and Embeddings with Trivial Normal (Conormal) Bundle

For a fixed $(n+1)$-manifold $Z$, the paper of immersions and embeddings $\beta: M \to Z$ with trivial normal line bundle leads to the construction of the sets $\mathsf{IMM}(Z)$ and $\mathsf{EMB}(Z)$, which classify quasitopy classes of such immersions and embeddings, respectively. The quasitopy relation is a hybrid of pseudo-isotopy and classical bordism: two immersions $\beta_0, \beta_1$ with trivialized normal bundles are quasitopic if there exists a compact $(n+1)$-manifold $N$ with boundary $(-M_0) \sqcup M_1$ and an immersion $B: N \to Z \times [0,1]$ with trivial normal bundle extending the given trivializations (Katz, 2023).

This equivalence relation gives rise to the following structure:

• $\mathsf{IMM}(Z)$: Quasitopy classes of immersions with trivial normal bundle.
• $\mathsf{EMB}(Z)$: Quasitopy classes of embeddings with trivial normal bundle.

A natural map $A: \mathsf{EMB}(Z) \to \mathsf{IMM}(Z)$, given by forgetting the "no self-intersections" condition, is injective and admits a right-inverse $R: \mathsf{IMM}(Z) \to \mathsf{EMB}(Z)$, built via a canonical resolution of self-intersections while preserving the triviality of the normal bundle. This correspondence allows classification and identification of obstructions for representing an immersion by an embedding (Katz, 2023).

3. Bordism Invariants and the Self-Intersection Map

The difference between immersions and embeddings with trivial conormal bundle is measured via the self-intersection bordism invariant. Given a $k$-normal immersion $\beta: M \to Z$ with trivial normal, the $k$-fold self-intersection locus

$$\Sigma_k(\beta) = (\beta^k)^{-1}(\Delta_Z) \subset M^k$$

is a smooth $(n - k + 1)$–manifold, and the projection $p_1 : \Sigma_k \to M \to Z$ defines a bordism class in $B_{n+1-k}(Z)$. Collecting these for $k = 2, \ldots, n+1$ assembles the map

$$\mathcal{B}\Sigma : \mathsf{IMM}(Z) \rightarrow \bigoplus_{k=2}^{n+1} B_{n+1-k}(Z),$$

which vanishes on $\mathsf{EMB}(Z)$. Consequently, $\mathcal{B}\Sigma$ yields an injection

$$\mathsf{IMM}(Z) / A(\mathsf{EMB}(Z)) \hookrightarrow \bigoplus_{k=2}^{n+1} B_{n+1-k}(Z),$$

so nontrivial self-intersection invariants are obstructions to quasitoping a trivial-normal immersion to an embedding (Katz, 2023).

4. Examples and Computational Aspects

Several explicit cases illustrate the structure of immersions and embeddings with trivial conormal bundle:

• For $Z = S^{n+1}$ or $\mathbb{R}^{n+1}$, where $H^1(Z) = 0$ and thus $\mathsf{EMB}(Z) = 0$, all trivial-normal immersions are nontrivial in quasitopy due to unattainable self-intersection invariants; $\mathsf{IMM}(Z) = B_n(Z)$ in this case (Katz, 2023).
• For $Z = T^3$ (the 3-torus), $$\mathsf{EMB}(T^3) \cong H^1(T^3) \cong \mathbb{Z}^3$$ and $$\mathsf{IMM}(T^3)/A(\mathsf{EMB}(T^3)) \cong B_1(T^3) \cong \mathbb{Z}^3$$. The canonical immersion of coordinate tori meeting at a single triple point represents a nontrivial class in the quotient, reflected in the odd 3-fold self-intersection number (Katz, 2023).
• In minimal-volume problems, if $\beta : M \rightarrow Z$ with trivial normal minimizes volume in its homology class, then $\beta$ must be an embedding. Any non-embedded immersion with trivial normal bundle can be resolved (via local surgeries preserving triviality), reducing volume strictly in each step (Katz, 2023).

5. Trivial Conormal Bundles in Legendrian and Contact Topology

In contact topology, the unit conormal bundle $\Lambda_K$ of $K \subset Q$ is a Legendrian submanifold of the unit cotangent bundle $S^* Q$. When the normal bundle $T K^\perp$ of $K$ is trivial, especially for codimension $d=2$, $\Lambda_K \cong K \times S^1$ admits a standard contact structure. In this case, various algebraic invariants simplify:

• The string topology model $H^{\mathrm{string}}_*(Q, K)$ is a nonnegatively graded $R$-algebra, with concrete computations in degree zero (the "cord algebra") directly reflecting the topology of $K$, and generators corresponding to homotopy classes of noncontractible "cords" in $Q \setminus K$ (Okamoto, 2022).
• In codimension two, the degree zero piece $H^{\mathrm{string}}_0(Q,K)$ is freely generated by homotopy classes of cords subject to explicit linear relations, and is identified with the cord algebra as in Ng–CELN for knots. The explicit realization of $H^{\mathrm{string}}_*(Q,K)$ in the trivial conormal case enables combinatorial computations (Okamoto, 2022).
• There is a conjectural isomorphism, supported by computations in low degrees and explicit examples (Hopf link vs unlink), between the string topology model and the Legendrian contact homology $LCH_*(\Lambda_K; R)$ when $Q = \mathbb{R}^n$ (Okamoto, 2022).

6. Technical Tools and Further Constructions

Triviality of the conormal (or normal) bundle allows significant technical simplifications:

• Global triviality enables a coherent choice of “positive chamber” in resolving self-intersections, reducing the topological complexity of local deformations in the ambient manifold.
• Thom's transversality and density results provide that $k$-normal immersions are dense, so any immersion can be approximated by one with only controlled transverse self-intersections, making quasitopy analysis tractable (Katz, 2023).

In contact topology, trivial conormal bundles admit a product description and standard contact structures, facilitating explicit algebraic and geometric models of unit conormals and their associated invariants.

7. Significance and Broader Context

The condition of trivial (co)normal bundle is central in both geometric topology and contact topology, underpinning classification of immersions and embeddings, construction of quantitative invariants (self-intersection bordism classes), and the development of algebraic invariants in string topology and Legendrian contact homology. The trivial conormal setting provides the foundational case for explicit computations, minimal-volume embeddings, and for the identification of topological and contact-topological invariants by combinatorial and algebraic means. This line of work unifies the paper of differentiable, topological, and symplectic structures, delineating the boundary between immersability, embeddability, and the richness of Legendrian and string-topological phenomena (Katz, 2023, Okamoto, 2022).