Interleaves

Many important spaces in topology and algebraic geometry have no odd-dimensional homology. For such spaces, functorial spatial homology truncation simplifies considerably. On the theory side, the simplification arises as follows: To define general spatial homology truncation, we used intermediate auxiliary structures, the n-truncation structures. For spaces that lack odd-dimensional homology, these structures can be replaced by a much simpler structure. Again every such space can be embedded in such a structure, which is the analogon of the general theory. On the application side, the crucial simplification is that the truncation functor t_<n will not require that in truncating a given continuous map, the map preserve additional structure on the domain and codomain of the map. In general, t_<n is defined on the category CW_n⊃∂, meaning that a map must preserve chosen subgroups “Y ”. Such a condition is generally necessary on maps, for otherwise no truncation exists. So arbitrary continuous maps between spaces with trivial odd-dimensional homology can be functorially truncated. In particular the compression rigidity obstructions arising in the general theory will not arise for maps between such spaces.

Let ICW be the full subcategory of CW whose objects are simply connected CW-complexes K with finitely generated even-dimensional homology and vanishing odd-dimensional homology for any coefficient group. We call ICW the interleaf category.

For example, the space K = S² ∪₂ e³ is simply connected and has vanishing integral homology in odd dimensions. However, H₃(K;Z/₂) = Z/₂ ≠ 0.

Let X be a space whose odd-dimensional homology vanishes for any coefficient group. Then the even-dimensional integral homology of X is torsion-free.

Taking the coefficient group Q/Z, we have

Tor(H_2k(X),Q/Z) = H_2k+1(X) ⊗ Q/Z ⊕ Tor(H_2k(X),Q/Z) = H_2k+1(X;Q/Z) = 0.

Thus H_2k(X) is torsion-free, since the group Tor(H_2k(X),Q/Z) is isomorphic to the torsion subgroup of H_2k(X).

Any simply connected closed 4-manifold is in ICW. Indeed, such a manifold is homotopy equivalent to a CW-complex of the form

V_i=1^kS_i²∪_ƒe⁴

where the homotopy class of the attaching map ƒ : S³ → V_i=1^k S_i² may be viewed as a symmetric k × k matrix with integer entries, as π₃(V_i=1^kS_i²) ≅ M(k), with M(k) the additive group of such matrices.

Any simply connected closed 6-manifold with vanishing integral middle homology group is in ICW. If G is any coefficient group, then H₁(M;G) ≅ H₁(M) ⊗ G ⊕ Tor(H₀M,G) = 0, since H₀(M) = Z. By Poincaré duality,

0 = H₃(M) ≅ H₃(M) ≅ Hom(H₃M,Z) ⊕ Ext(H₂M,Z),

so that H₂(M) is free. This implies that Tor(H₂M,G) = 0 and hence H₃(M;G) ≅ H₃(M) ⊗ G ⊕ Tor(H₂M,G) = 0. Finally, by G-coefficient Poincaré duality,

H₅(M;G) ≅ H₁(M;G) ≅ Hom(H₁M,G) ⊕ Ext(H₀M,G) = Ext(Z,G) = 0

Any smooth, compact toric variety X is in ICW: Danilov’s Theorem implies that H^∗(X;Z) is torsion-free and the map A^∗(X) → H^∗(X;Z) given by composing the canonical map from Chow groups to homology, A^k(X) = A_n−k(X) → H_2n−2k(X;Z), where n is the complex dimension of X, with Poincaré duality H_2n−2k(X;Z) ≅ H_2k(X;Z), is an isomorphism. Since the odd-dimensional cohomology of X is not in the image of this map, this asserts in particular that H_odd(X;Z) = 0. By Poincaré duality, H_even(X;Z) is free and H_odd(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that H_odd(X;G) = 0 for any coefficient group G. If we only wanted to establish H_odd(X;Z) = 0, then it would of course have been enough to know that the canonical, degree-doubling map A^∗(X) → H^∗(X;Z) is onto. One may then immediately reduce to the case of projective toric varieties because every complete fan Δ has a projective subdivision Δ^′, the corresponding proper birational morphism X(Δ^′) → X(Δ) induces a surjection H^∗(X(Δ^′);Z) → H^∗(X(Δ);Z) and the diagram

commutes.

Let G be a complex, simply connected, semisimple Lie group and P ⊂ G a connected parabolic subgroup. Then the homogeneous space G/P is in ICW. It is simply connected, since the fibration P → G → G/P induces an exact sequence

1 = π₁(G) → π₁(G/P) → π₀(P) → π₀(G) = 0,

which shows that π₁(G/P) → π₀(P) is a bijection. Accordingly, ∃ elements s_w(P) ∈ H_2l(w)(G/P;Z) (“Schubert classes,” given geometrically by Schubert cells), indexed by w ranging over a certain subset of the Weyl group of G, that form a basis for H_∗(G/P;Z). (For w in the Weyl group, l(w) denotes the length of w when written as a reduced word in certain specified generators of the Weyl group.) In particular H_even(G/P;Z) is free and H_odd(G/P;Z) = 0. Thus H_odd(G/P;G) = 0 for any coefficient group G.

The linear groups SL(n, C), n ≥ 2, and the subgroups S p(2n, C) ⊂ SL(2n, C) of transformations preserving the alternating bilinear form

x₁y_n+1 +···+ x_ny_{2n −xn+1y1 −···−x2nyn}

on C²ⁿ × C²ⁿ are examples of complex, simply connected, semisimple Lie groups. A parabolic subgroup is a closed subgroup that contains a Borel group B. For G = SL(n,C), B is the group of all upper-triangular matrices in SL(n,C). In this case, G/B is the complete flag manifold

G/B = {0 ⊂ V₁ ⊂···⊂ V_n−1 ⊂ C_n}

of flags of subspaces V_i with dimV_i = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a half-flag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a Grassmannian G(k, n) of k-dimensional subspaces of Cⁿ. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic k-dimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming fibrations.

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The Illicit Trade of Firearms, Explosives and Ammunition on the Dark Web

DATACRYPTO is a web crawler/scraper class of software that systematically archives websites and extracts information from them. Once a cryptomarket has been identified, DATACRYPTO is set up to log in to the market and download its contents, beginning at the web page fixed by the researchers (typically the homepage). After downloading that page, DATACRYPTO parses it for hyperlinks to other pages hosted on the same market and follows each, adding new hyperlinks encountered, and visiting and downloading these, until no new pages are found. This process is referred to as web crawling. DATACRYPTO then switches from crawler to scraper mode, extracting information from the pages it has downloaded into a single database.

One challenge connected to crawling cryptomarkets arises when, despite appearances to the contrary, the crawler has indexed only a subset of a marketplace’s web pages. This problem is particularly exacerbated by sluggish download speeds on the Tor network which, combined with marketplace downtime, may prevent DATACRYPTO from completing the crawl of a cryptomarket. DATACRYPTO was designed to prevent partial marketplace crawls through its ‘state-aware’ capability, meaning that the result of each page request is analysed and logged by the software. In the event of service disruptions on the marketplace or on the Tor network, DATACRYPTO pauses and then attempts to continue its crawl a few minutes later. If a request for a page returns a different page (e.g. asking for a listing page and receiving the home page of the cryptomarket), the request is marked as failed, with each crawl tallying failed page requests.

DATACRYPTO is programmed for each market to extract relevant information connected to listings and vendors, which is then collected into a single database:

• Product title;
• Product description;
• Listing price;
• Number of customer feedbacks for the listing;
• The country or region from which a vendor ships the product;
• The country or regions to which the vendor placing the listing is willing to ship.

DATACRYPTO is not the first crawler to mirror the dark web, but is novel in its ability to pull information from a variety of cryptomarkets at once, despite differences in page structure and naming conventions across sites. For example, “$…” on one market may give you the price of a listing. On another market, price might be signified by “VALUE…” or “PRICE…” instead.

Researchers who want to create a similar tool to gather data through crawling the web should detail which information exactly they would like to extract. When building a web crawler it is, for example, very important to carefully study the structure and characteristics of the websites to be mirrored. Before setting the crawler loose, ensure that it extracts and parses correct and complete information. Because the process of building a crawler-tool like DATACRYPTO can be costly and time consuming, it is also important to anticipate on future data needs, and build in capabilities to extract that kind of data later on, so no large future modifications are necessary.

Building a complex tool like DATACRYPTO is no easy feat. The crawler needs to be able to copy pages, but also stealthily get around CAPTCHAs and log itself in onto the TOR server. Due to their bulkiness, web crawlers can place a heavy burden on a website’s server, and are easily detected due to their repetitive pattern moving between pages. Site administrators are therefore not afraid to IP-ban badly designed crawlers from their sites.

The Illicit Trade of Firearms Explosives and Ammunition on the Dark Web

Collateral Debt Obligations. Thought of the Day 111.0

A CDO is a general term that describes securities backed by a pool of fixed-income assets. These assets can be bank loans (CLOs), bonds (CBOs), residential mortgages (residential- mortgage–backed securities, or RMBSs), and many others. A CDO is a subset of asset- backed securities (ABS), which is a general term for a security backed by assets such as mortgages, credit card receivables, auto loans, or other debt.

To create a CDO, a bank or other entity transfers the underlying assets (“the collateral”) to a special-purpose vehicle (SPV) that is a separate legal entity from the issuer. The SPV then issues securities backed with cash flows generated by assets in the collateral pool. This general process is called securitization. The securities are separated into tranches, which differ primarily in the priority of their rights to the cash flows coming from the asset pool. The senior tranche has first priority, the mezzanine second, and the equity third. Allocation of cash flows to specific securities is called a “waterfall”. A waterfall is specified in the CDO’s indenture and governs both principal and interest payments.

1: If coverage tests are not met, and to the extent not corrected with principal proceeds, the remaining interest proceeds will be used to redeem the most senior notes to bring the structure back into compliance with the coverage tests. Interest on the mezzanine securities may be deferred and compounded if cash flow is not available to pay current interest due.

One may observe that the creation of a CDO is a complex and costly process. Professionals such as bankers, lawyers, rating agencies, accountants, trustees, fund managers, and insurers all charge considerable fees to create and manage a CDO. In other words, the cash coming from the collateral is greater than the sum of the cash paid to all security holders. Professional fees to create and manage the CDO make up the difference.

CDOs are designed to offer asset exposure precisely tailored to the risk that investors desire, and they provide liquidity because they trade daily on the secondary market. This liquidity enables, for example, a finance minister from the Chinese government to gain exposure to the U.S. mortgage market and to buy or sell that exposure at will. However, because CDOs are more complex securities than corporate bonds, they are designed to pay slightly higher interest rates than correspondingly rated corporate bonds.

CDOs enable a bank that specializes in making loans to homeowners to make more loans than its capital would otherwise allow, because the bank can sell its loans to a third party. The bank can therefore originate more loans and take in more origination fees. As a result, consumers have more access to capital, banks can make more loans, and investors a world away can not only access the consumer loan market but also invest with precisely the level of risk they desire.

1: To the extent not paid by interest proceeds.

2: To the extent senior note coverage tests are met and to the extent not already paid by interest proceeds. If coverage tests are not met, the remaining principal proceeds will be used to redeem the most senior notes to bring the structure back into compliance with the coverage tests. Interest on the mezzanine securities may be deferred and compounded if cash flow is not available to pay current interest due.

The Structured Credit Handbook provides an explanation of investors’ nearly insatiable appetite for CDOs:

Demand for [fixed income] assets is heavily bifurcated, with the demand concentrated at the two ends of the safety spectrum . . . Prior to the securitization boom, the universe of fixed-income instruments issued tended to cluster around the BBB rating, offering neither complete safety nor sizzling returns. For example, the number of AA and AAA-rated companies is quite small, as is debt issuance of companies rated B or lower. Structured credit technology has evolved essentially in order to match investors’ demands with the available profile of fixed-income assets. By issuing CDOs from portfolios of bonds or loans rated A, BBB, or BB, financial intermediaries can create a larger pool of AAA-rated securities and a small unrated or low-rated bucket where almost all the risk is concentrated.

CDOs have been around for more than twenty years, but their popularity skyrocketed during the late 1990s. CDO issuance nearly doubled in 2005 and then again in 2006, when it topped $500 billion for the first time. “Structured finance” groups at large investment banks (the division responsible for issuing and managing CDOs) became one of the fastest-growing areas on Wall Street. These divisions, along with the investment banking trading desks that made markets in CDOs, contributed to highly successful results for the banking sector during the 2003–2007 boom. Many CDOs became quite liquid because of their size, investor breadth, and rating agency coverage.

Rating agencies helped bring liquidity to the CDO market. They analyzed each tranche of a CDO and assigned ratings accordingly. Equity tranches were often unrated. The rating agencies had limited manpower and needed to gauge the risk on literally thousands of new CDO securities. The agencies also specialized in using historical models to predict risk. Although CDOs had been around for a long time, they did not exist in a significant number until recently. Historical models therefore couldn’t possibly capture the full picture. Still, the underlying collateral could be assessed with a strong degree of confidence. After all, banks have been making home loans for hundreds of years. The rating agencies simply had to allocate risk to the appropriate tranche and understand how the loans in the collateral base were correlated with each other – an easy task in theory perhaps, but not in practice.

The most difficult part of valuing a CDO tranche is determining correlation. If loans are uncorrelated, defaults will occur evenly over time and asset diversification can solve most problems. With low correlation, an AAA-rated senior tranche should be safe and the interest rate attached to this tranche should be close to the rate for AAA-rated corporate bonds. High correlation, however, creates nondiversifiable risk, in which case the senior tranche has a reasonable likelihood of becoming impaired. Correlation does not affect the price of the CDO in total because the expected value of each individual loan remains the same. Correlation does, however, affect the relative price of each tranche: Any increase in the yield of a senior tranche (to compensate for additional correlation) will be offset by a decrease in the yield of the junior tranches.