COMPLEX BORDISM and RELATED SINGULAR BORDISM Contents 1

COMPLEX BORDISM AND RELATED SINGULAR BORDISM

JIAHAO HU

Abstract. In this note, we discuss how to build geometric homology theories using manifolds with singularities.

Contents 1. Introduction 1 2. Stably almost complex manifolds and complex bordism theory 2 2.1. Stably almost complex manifolds 2 2.2. Complex bordism ring 2 2.3. Complex bordism theory 3 3. Join-like singularities and singular bordism theories 3 3.1. Join-like singularities 3 3.2. Singular bordism theories 4 4. Singular homology as a singular bordism theory 5 4.1. Baas-Sullivan exact sequence 5 4.2. Singular bordism of a point 6 5. Final remarks 6 5.1. Representation of singular homology class 6 5.2. Other interesting geometric theories 6 References 6

1. Introduction What do homology classes look like? From definition, all singular homology classes are represented by singular geometric cycles, which can carry all kinds of singularities. It is natural to ask, can we find better representatives? Of course, the best kind of representatives will be smooth manifolds–no singularities at all. Definition 1.1. We say a homology class is representable by manifolds if it is a push-forward of fundamental class of a closed oriented manifold. It is not hard to see that all 0, 1, 2-dimensional homology classes are represented by manifolds. In fact, thanks to Thom, all homology classes up to dimension 6 is representable by manifolds. However, Thom also found a counterexample in dimension 7.

Example 1.2. There is a homology class x1 ∗ x5 ∈ H7(K(Z/3 × Z/3, 1); Z) that is not representable by manifolds. 1 2 JIAHAO HU

So there are inately singular homology classes, which means our hope to represent homology classes by manifolds fails. Here’s where Dennis Sullivan came along and pointed out that, even though we can not always represent homology classes by manifolds, we can still represent them by manifolds with singularities, more precisely by stably almost complex manifolds with prescribed type of singularities. In order to explain this point of Dennis Sullivan, this note is structured as following. We first discuss stably almost complex manifolds and related smooth bordism theory. Then we throw in certain a-priori described singularities into the bordism picture and form singular bordism theories. Finally, we discuss a long exact sequence relating different singular bordism theories, and thus prove the representablity statement of Sullivan. Without further notice, all homology groups in this note will have integral coefficients and all manifolds are assumed to be smooth and closed.

2. Stably almost complex manifolds and complex bordism theory 2.1. Stably almost complex manifolds. Let’s recall the deﬁnitions of almost complex and stably almost complex manifolds.

Definition 2.1. A manifold M is almost complex if its tangent bundle TM is a complex vector bundle, i.e. there is an endomorphism J : TM → TM such that J 2 = −1. Definition 2.2. A manifold M is stably almost complex (or simply SAC) if T ⊕εn M R is a complex vector bundle for some n, where εR is the trivial bundle M × R → M. All almost complex manifolds are even dimensioanl and orientable, while stably almost complex manifolds are orientable but can be odd dimensional. For example, all odd spheres are stably almost complex. If (M,J) is SAC, then −M (M with opposite orientation) is also SAC endowed with −J. We view M and −M as different SAC manifolds. 2.2. Complex bordism ring. We now proceed discussing bordism theories, where manifolds with boundaries are central objects. So we will stick to the category of stably almost complex manifolds instead of almost complex manifolds, for simple dimension reason. Definition 2.3. A SAC manifold with boundary is an oriented manifold with boundary (N, ∂N) such that both TN and T∂N have SAC structures and the SAC structure on T∂N is the one restricted from TN . (Notice T∂N ⊕ εR = TN .)

Definition 2.4. Let M1,M2 be two SAC manifolds of same dimension. We say M1 is SAC bordant to M2 if M1 ∪ (−M2) = ∂N where N is a SAC manifold with boundary. Now we can define complex bordism groups. U Definition 2.5. Let Ωn be the set of n-dimensional SAC manifolds modulo SAC bordism. We justify the name bordism group by the following observation. Proposition 2.6. Under the natural operations disjoint union and Cartisian prod- U uct, Ω∗ forms a graded (commutative) ring (with unit), with zero being the empty set and unit being a single point. COMPLEX BORDISM AND RELATED SINGULAR BORDISM 3

With a huge effort, one can compute the complex bordism ring. Theorem 2.7 (Milnor-Novikov-Quillen-Thom). U Ω∗ ' Z[x2, x4, x6 ... ] is a polynomial ring generated by elements in degree 2, 4, 6,... and U 1 2 3 Ω∗ ⊗Z Q ' Q[CP , CP , CP ,... ]. Remark 2.8. Complex projective spaces are not integral generators. 2.3. Complex bordism theory. We can extend the notion of bordism to maps and get complex bordism theory. Definition 2.9. Let (X,A) be a pair of finite CW complexes, then the n-th com- U plex bordism group of pair (X,A) is defined to be the set Ωn (X,A) consisting of all continuous maps (N, ∂N) → (X,A), where (N, ∂N) is a n-dimensional SAC manifold with boundary, modulo obvious SAC bordism of maps.

U U It is obvious that Ωn (X,A) is a group under disjoint union, and Ω∗ (X,A) is U a graded module over the complex bordism ring Ω∗ . U Theorem 2.10. Ω∗ (−, −) is a generalised homology theory, i.e. it satiﬁes all the U U Eilenberg-Steenrod axioms except for the dimension axiom. In fact, Ω∗ (pt) = Ω∗ .

3. Join-like singularities and singular bordism theories In this section, we follow Sullivan’s treatment on join-like singularities in [1]. Instead of giving rigorous deﬁnitions (which means I will lie at some point), we illustrate the idea by examples. For a rigorous detailed discussion, see [2]. 3.1. Join-like singularities. Consider any sequence of distinct closed (SAC) manifolds M1,M2, ··· ,Mi, ··· .

If I = (i0, . . . , ir) is any ﬁnite set of distinct indices, consider the join

MI = Mi0 ∗ Mi2 ∗ · · · ∗ Mir .

Recall that the points of MI are the points of all possible r-simplicies whose vertices lie (respectively) in the disjoint union Mi0 ∪ Mi1 ∪ · · · ∪ Mir . Example 3.1. S1 ∗ Z/4 = S1 × Cone(Z/4). Two 2-spheres intersecting each other in R3 has this type of singularity. n Example 3.2. Consider S and identify q distinct points x1, . . . xq, the quotient space n S /{x1 = ··· = xq} has singularity type pt ∗ (Sn−1 × Z/q) = Cone(Sn−1 × Z/q). Example 3.3. Consider Sk ∗ M m, we examine its local structure. If p ∈ Sk, then it has a neighborhood homeomorphic to D1 × Cone(M). If p ∈ M, then it has a neighborhood homeomorphic to Cone(Sk) × Dm ' Dk+1 × Dm ' Dk+m+1. If p is somewhere in between, then it has a neighborhood homeomorphic to Dk ∗ Dm ' Dk+m+1. So Sk ∗ M is smooth except for Sk, and its singularity looks like D1 × Cone(M). 4 JIAHAO HU

In general, if we choose Mi0 to be some positive dimensional sphere, then MI is singular at those boundary points of the simplex not in the open face opposite the vertex in Mi0 . And each point p in MI has a neighborhood homeomorphic to

(Euclidean space) × Cone(MJ ) with J ⊂ (i0, i1, . . . , ir). J is the set of indicies for which the natural barycentric coordinates of p (excluding i0) vanish. Definition 3.4. We say a stratified space W is a manifold with join-like singularities with respect to the sequence S = {M0 = pt, M1,...,Mi,... } (or simply a S-manifold) if it locally looks like (Euclidean space) × Cone(MI ), and (technical condition) the singularities are assembled together in a coherent way. Remark 3.5. Roughly a stratified space is a manifold with singularity and its singularity is again a manifold with singularity and so on so forth. So we can make sense of dimension of a stratified space. Remark 3.6. Any manifold with boundary (M, ∂M) is a stratified space, whose singularity is ∂M and locally looks like (Euclidean space) × Cone(pt). Example 3.7. Smooth manifolds have no singularities, hence are manifolds with join-like singularities.

Example 3.8. In particular W = Cone(Mi) is a stratified space with join-like singularities with respect to {Mi}, so is (smooth space) × Cone(Mi). 3.2. Singular bordism theories. Now we fix a sequence of (SAC) manifolds S = {M0 = pt, M1,M2,... } and build a singular bordism theory using S-manifolds. We have to make sense of boundary first. Definition 3.9. Suppose W is a S-manifold, then we define ∂W to be the set containing all points p ∈ M such that a neighborhood of p is homeomorphic to

(Eulclidean space) × Cone(M0 ∗ MJ ). Remark 3.10. This is the point where I start to lie, and all the following deﬁnitions related to the above one. Again for a rigorous deﬁnition, see [2].

Example 3.11. ∂(Cone(Mi)) = Mi. Example 3.12. Two 2-sphere intersecting in R3 has no boundary. We note that under this deﬁnition, it is clear that ∂W is also a S-manifold. Indeed, any point p ∈ ∂W has neighborhood homeomorphic to

(Euclidean space) × Cone(MJ ) in ∂W and the smooth loci is where the neighborhood in W has form

(Euclidean space) × Cone(M0) ' (Euclidean space) × [0, 1]. It follows that ∂W is of one dimension lower than W . Similar to the smooth case, we can now form singular bordism groups U Ω∗ (S)(X,A) containing maps from S-manifolds (W, ∂W ) → (X,A) modulo obvious bordism equivalence. They are abelian groups under disjoint union, and graded module U over Ω∗ . COMPLEX BORDISM AND RELATED SINGULAR BORDISM 5

U Theorem 3.13 (Baas-Sullivan). For any such a sequence S, Ω∗ (S)(−, −) is a homology theory. This provides a happy family of geometric homology theories. But this abstract construction is somewhat obscure so far, for instance we do not even know U Ω∗ (S)(pt). U From definition Ω∗ (S)(pt) contains S-manifolds modulo bordism. But there’s no effective way to compute it. If one is familiar with smooth bordism theory, in the smooth case we can turn this problem into a computation of homotopy groups of certain spaces (Thom space), but that method doesn’t seem to work here since there’s no good description of the corresponding spaces. We need to find another route. U To compute Ω∗ (S), we first have the following observation. U Proposition 3.14. There is a natural Ω∗ -module map U U Ω∗ → Ω∗ (S). U such that I(S), the ideal of Ω∗ generated by S − {M0}, is contained the kernel of this map. Indeed, Mi bounds Cone(Mi) as S-manifold for i > 0. U U So there is an induced map Ω∗ /I(S) → Ω∗ (S). To understand this map better, we need a exact sequence discussed in the next section.

4. Singular homology as a singular bordism theory In this section, we develope an exact sequence, which allows us to inductively compute the singular bordism groups of a point.

4.1. Baas-Sullivan exact sequence. Let Sn = {M0,M1,...,Mn}, we build an U U U U exact sequence relating Ω∗ (Sn) and Ω∗ (Sn+1). Note that Ω∗ (S0) = Ω∗ , thus this U will allow us to inductively compute Ω∗ (S) (under good situations). To start with, we define several homomorphisms. Let X be a finite CW complex. (1) Let U U ×[Mn+1]:Ω∗ (Sn)(X) → Ω∗ (Sn)(X) be defined by

(W → X) 7→ (W × Mn+1 → W → X).

×[Mn+1] is obvously well-deﬁned and of degree + dim Mn+1. (2) Let U U γ :Ω∗ (Sn)(X) → Ω∗ (Sn+1)(X) be the natural map induced by viewing Sn-manifolds as Sn+1-manifolds. γ is of degree 0. (3) Let U U δ :Ω∗ (Sn+1)(X) → Ω∗ (Sn)(X) 0 be the restriction map from a Sn+1 manifold W to its subspace W consisting of point whose local neighborhood is homeomorphic to

(Euclidean space) × Cone(MI ) 0 where Mi ∈ Sn for all i ∈ I. Trivially W is a Sn-manifold by deﬁnition. δ is of degree − dim Mn+1 − 1. (For instance, Cone(Mn+1) is viewed as empty set.) 6 JIAHAO HU

Theorem 4.1. The following sequence is exact

U ×[Mn+1] U γ U δ · · · → Ω∗ (Sn)(X) −−−−−−→ Ω∗ (Sn)(X) −→ Ω∗ (Sn+1)(X) −→· · · . 4.2. Singular bordism of a point. We apply the above sequence to compute singular bordism of a point.

Theorem 4.2. Let S = {M0 = pt, M1,M2,... } be a sequence of SAC manifolds. U Suppose M1,M2,...,Mi,... form a regular sequence of complex bordism ring Ω∗ , then U U Ω∗ (S) ' Ω∗ /I(S). U U Proof. We prove by induction. It is clear Ω∗ (S0) = Ω∗ . Suppose inductively we al- U U ready have Ω∗ (Sn) ' Ω∗ /(M1,...,Mn), by the assumption that M1,...,Mn,Mn+1 is a regular sequence, the map ×[Mn+1] is injective. Therefore, we have short exact sequence U ×[Mn+1] U γ U δ 0 → Ω∗ (Sn) −−−−−−→ Ω∗ (Sn) −→ Ω∗ (Sn+1) −→ 0 and the theorem follows. U Corollary 4.3. If we choose S to be a set of regular generator of Ω∗ , S = U {x2, x4,... } for instance, then Ω∗ (S)(−, −) is the singular homology theory. U Proof. In this case, Ω∗ (S) ' Z and the corollary follows from Eilenberg-Steenrod uniqueness of singular homology theory. 5. Final remarks 5.1. Representation of singular homology class. So every singular homology class can be represented by SAC manifold with join-like singularities with respect U to the sequence {x2, x4,... }. Note though the choice of generators of Ω∗ is not canonical, hence our representation is not canonical. 5.2. Other interesting geometric theories. If one choose S to be the set of regular generators of the ideal generated by zero Todd genus manifolds, then one obtains a version of connective K-theory. Real K-theory (away from 2), elliptic homology (connective version) can be constructed in a similar way. The Baas-Sullivan exact sequence and the discussion of regular sequence leads to the well celebrated Landweber exactness theorem, from which one can construct a zoo of homology theories associated to genera.

References [1] D. Sullivan. Singularities in spaces. Proc. of Liverpool Singularities Symposium II, Lecture Notes in Mathematics 209, pp. 196-206, (1971). [2] N. A. Baas. On bordism theory of manifolds with singularities. Mathematica Scandinavica, 33, 279-302, (1973).