By Vilgelm Ilich Fushchich, A. G. Nikitin, W. I. Fushchich

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**Example text**

Consider the geometric boundary eﬀect TGSS diﬀerential 2 dS (η [4]) = η 3 [8, 2]. 2 gives a TEHPSS diﬀerential dTEHP (η [4]) = η 3 [8, 2]. 6. 7). Consider the bizarre diﬀerential dGSS (θ3 [4]) = 1[15, 3] in the TGSS for S 1 . 2 gives a corresponding TEHPSS diﬀerential dTEHP (θ3 [4]) = 1[15, 3]. 4. 2. 1. This “rogue diﬀerential” is given by the following lemma. 1. There is a non-trivial TEHPSS diﬀerential dTEHP (η 2 [13, 6]) = ηα8/5 [3]. 4. A BAD DIFFERENTIAL 51 Proof. 2, that η 2 [13, 6] is in the kernel of the TEHPSS diﬀerentials ∗ ∗ → E∗,[J,5] dTEHP : E∗,[6,2] for all J.

Diﬀerentials in the TAHSS, as in any Atiyah-Hirzebruch-type spectral sequence, can be eﬀectively computed from a sound understanding of the attaching map structure in the CW-spectrum L(k)n . The cells e J associated to CU sequences J are in bijective correspondence with the homology elements [J] ∈ H∗ (L(k)n ). Often, these attaching maps can be determined from the action of the dual Steenrod operations on ¯n H∗ (L(k)n ) ∼ =R given by the Nishida relations. This will be our main technique for determining TAHSS diﬀerentials in the sample calculations of Chapter 6.

Suppose that β[m] is the target of a shorter dS -diﬀerential. The only possibility is that β[m] is the target of a diﬀerential in the AHSS for L(1)n . Since the AHSS for L(1)n is a truncation of the AHSS for L(1)1 , this would imply that β[m] is the target of a diﬀerential in the AHSS for L(1)1 . This is impossible, as β[m] detects JH(α) = 0. 2) must be non-trivial. Suppose that m < n. Then we conclude that α is in the kernel of the diﬀerential ∗ ∗ (S n ) → E∗,[m] (S n ). dS : E∗,∅ n In particular, α is in the kernel of the shorter diﬀerentials ∗ ∗ dS : E∗,∅ (S n ) → E∗,[j] (S n ), n j ≥ n.