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@article{Husemoller1993,
abstract = {manifold. The prototype example for our discussion will be the m{\"{o}}bius band, as it is the simplest example of a nontrivial fiber bundle. We can create the m{\"{o}}bius band by starting with the circle S¹ and (similarly to the case with the cylinder) at each point on the circle attaching a copy of the open interval (0,1), but in a nontrivial manner. Instead of just attaching a bunch of parallel intervals to the circle, our intervals perform a 180° twist as we go around. This gives the manifold a much more interesting geometry. We look at the object we've formed, and note that locally it is indistinguishable from the cylinder. That is, the " twist " in the m{\"{o}}bius band is not located at any particular point on the band; it is entirely a global property of the manifold. Motivated by this example, we seek to generalize the language of product spaces, to include objects like the m{\"{o}}bius band which are only locally a product space. This generalization is what we will come to know as a fiber bundle. Figure 4.2 The mobius band can also be built from a circle and line segment, but using a more general method of construction.},
author = {Husemoller, Dale},
doi = {10.1016/B978-0-08-092521-9.50007-X},
file = {:home/zack/Dropbox/Mendeley/Husemoller/Unknown/Husemoller - 1993 - Fiber Bundles.pdf:pdf},
isbn = {9781475722635},
issn = {0072-5285},
number = {2},
pages = {368},
title = {{Fiber Bundles}},
volume = {1},
year = {1993}
}
@article{Wright,
abstract = {We define fiber bundles and discuss the long exact sequence of homotopy groups of a fiber bundle, and we give the Hopf bundles as examples. We also prove the Freudenthal suspen-sion theorem for spheres. All results are applied immediately to homotopy groups of spheres.},
author = {Wright, Alex},
file = {:home/zack/Dropbox/Mendeley/Wright/Unknown/Wright - Unknown - HOMOTOPY GROUPS OF SPHERES A VERY BASIC INTRODUCTION.pdf:pdf},
title = {{HOMOTOPY GROUPS OF SPHERES: A VERY BASIC INTRODUCTION}},
url = {https://web.stanford.edu/{~}amwright/HomotopyGroupsOfSoheres.pdf}
}
@article{Bubenik2017,
abstract = {Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning. We give efficient algorithms for calculating persistence landscapes, their averages, and distances between such averages. We discuss an implementation of these algorithms and some related procedures. These are intended to facilitate the combination of statistics and machine learning with topological data analysis. We present an experiment showing that the low-dimensional persistence landscapes of points sampled from spheres (and boxes) of varying dimensions differ.},
author = {Bubenik, Peter and D{\l}otko, Pawe{\l}},
doi = {10.1016/j.jsc.2016.03.009},
journal = {J. Symb. Comput.},
title = {{A persistence landscapes toolbox for topological statistics}},
url = {http://www.mendeley.com/research/persistence-landscapes-toolbox-topological-statistics},
year = {2017}
}
@article{May,
author = {May, J P and Ponto, K},
file = {:home/zack/Dropbox/Mendeley/May, Ponto/Unknown/May, Ponto - Unknown - More Concise Algebraic Topology Localization, completion, and model categories.pdf:pdf},
title = {{More Concise Algebraic Topology: Localization, completion, and model categories}},
url = {https://web.math.rochester.edu/people/faculty/doug/otherpapers/mayponto.pdf}
}
@article{Olah1996b,
author = {Olah et al},
file = {:home/zack/Dropbox/Mendeley/Olah et al/Unknown/Olah et al - Unknown - Spectral Sequence of Inclusion of a Subspace.pdf:pdf},
title = {{Spectral Sequence of Inclusion of a Subspace}}
}
@article{Weibel,
author = {Weibel, Charles},
file = {:home/zack/Dropbox/Mendeley/Weibel/Unknown/Weibel - Unknown - Projective Modules and Vector Bundles.pdf:pdf},
isbn = {9780821891322},
pages = {1--54},
title = {{Projective Modules and Vector Bundles}}
}
@article{Ultrametric1961,
abstract = {The landscape of homological algebra has evolved over the past half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras is also described. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors.},
archivePrefix = {arXiv},
arxivId = {arXiv:cond-mat/0104167},
author = {Ultrametric, W H Schikhof},
doi = {10.1016/0016-0032(61)90240-X},
eprint = {0104167},
file = {:home/zack/Dropbox/Mendeley/Ultrametric/Journal of the Franklin Institute/Ultrametric - 1961 - An introduction to homological algebra.pdf:pdf},
isbn = {0521435005;},
issn = {00160032},
journal = {J. Franklin Inst.},
keywords = {Homological Algebra, Mathematics},
number = {2},
pages = {152},
pmid = {10575050},
primaryClass = {arXiv:cond-mat},
title = {{An introduction to homological algebra}},
volume = {271},
year = {1961}
}
@article{Ramos2017a,
abstract = {These are lecture notes for a short course about spectral sequences that was held at M$\backslash$'alaga, October 18--20 (2016), during the "Fifth Young Spanish Topologists Meeting". The approach was to illustrate the basic notions via fully computed examples arising from Algebraic Topology and Group Theory.},
archivePrefix = {arXiv},
arxivId = {1702.00666},
author = {Ramos, Antonio D{\'{i}}az},
eprint = {1702.00666},
file = {:home/zack/Dropbox/Mendeley/Ramos/Unknown/Ramos - 2017 - Spectral sequences via examples.pdf:pdf},
pages = {1--31},
title = {{Spectral sequences via examples}},
url = {http://arxiv.org/abs/1702.00666},
year = {2017}
}
@article{Sharifi,
author = {Sharifi, Romyar},
file = {:home/zack/Dropbox/Mendeley/Sharifi/Unknown/Sharifi - Unknown - Abstract Algebra.pdf:pdf},
title = {{Abstract Algebra}}
}
@article{Schwede2008,
abstract = {The symmetric spectra introduced by Hovey, Shipley and Smith are a convenient model for the stable homotopy category with a nice associative and commutative smash product on the point set level and a compatible Quillen closed model structure. About the only disadvantage of this model is that the stable equivalences cannot be defined by inverting those morphisms which induce isomorphisms on homotopy groups, because this would leave too many homotopy types. In this sense the naively defined homotopy groups are often `wrong`, and then their precise relationship to the `true` homotopy groups (i.e., morphisms in the stable homotopy category from sphere spectra) appears mysterious. In this paper I discuss and exploit extra algebraic structure on the naively defined homotopy groups of symmetric spectra, namely a special kind of action of the monoid of injective self maps of the natural numbers. This extra structure clarifies several issues about homotopy groups and stable equivalences and explains why the naive homotopy groups are not so wrong after all. For example, the monoid action allows a simple characterization of semistable symmetric spectra.},
author = {Schwede, Stefan},
doi = {10.2140/gt.2008.12.1313},
journal = {Geom. Topol.},
title = {{On the homotopy groups of symmetric spectra}},
url = {http://www.mendeley.com/research/homotopy-groups-symmetric-spectra},
year = {2008}
}
@book{Bott,
author = {Bott, Raoul and Tu, Loring W.},
file = {:home/zack/Dropbox/Mendeley/Bott, Tu/Unknown/Bott, Tu - Unknown - Differential Forms in Algebraic Topology.pdf:pdf},
isbn = {0387906134},
title = {{Differential Forms in Algebraic Topology}}
}
@article{Stevens2009,
author = {Stevens, P F and Kaesuk, Carol and Norton, Yoon W W},
file = {:home/zack/Dropbox/Mendeley/Stevens, Kaesuk, Norton/Unknown/Stevens, Kaesuk, Norton - 2009 - An Essay on Spectral Sequences.pdf:pdf},
isbn = {9780393061970},
number = {1},
pages = {127--148},
title = {{An Essay on Spectral Sequences}},
volume = {100},
year = {2009}
}
@article{Complexes1998,
author = {Complexes, Simplicial},
file = {:home/zack/Dropbox/Mendeley/Complexes/Unknown/Complexes - 1998 - First Examples of Free Resolutions.pdf:pdf},
title = {{First Examples of Free Resolutions}},
year = {1998}
}
@article{Definitions,
author = {Definitions, Desiderata},
file = {:home/zack/Dropbox/Mendeley/Definitions/Unknown/Definitions - Unknown - Tensor Products.pdf:pdf},
title = {{Tensor Products}}
}
@article{Beke2010,
abstract = {There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan's Ex functor act on graphs and on nerves of small categories.},
author = {Beke, Tibor},
doi = {10.1007/s10485-009-9190-7},
journal = {Appl. Categ. Struct.},
title = {{Fibrations of simplicial sets}},
url = {http://www.mendeley.com/research/fibrations-simplicial-sets},
year = {2010}
}
@article{Freedman2009,
abstract = {Algebraic topology is generally considered one of the purest subfields of mathematics. However, over the last decade two interesting new lines of research have emerged, one focusing on algorithms for algebraic topology, and the other on applications of algebraic topology in engineering and science. Amongst the new areas in which the techniques have been applied are computer vision and image processing. In this paper, we survey the results of these endeavours. Because algebraic topology is an area of mathematics with which most computer vision practitioners have no experience, we review the machinery behind the theories of homology and persistent homology; our review emphasizes intuitive explanations. In terms of applications to computer vision, we focus on four illustrative problems: shape signatures, natural image statistics, image denoising, and segmentation. Our hope is that this review will stimulate interest on the part of computer vision researchers to both use and extend the tools of this new field},
author = {Freedman, Daniel and Chen, Chao},
journal = {Comput. Vis.},
title = {{Algebraic topology for computer vision}},
url = {http://www.mendeley.com/research/algebraic-topology-computer-vision},
year = {2009}
}
@article{2011,
author = {Island, Rhode},
file = {:home/zack/Dropbox/Mendeley/Island/Unknown/Island - Unknown - Proceedings of AT 1970.pdf:pdf},
title = {{Proceedings of AT 1970}}
}
@article{Lyczak1972,
author = {Nadim, Jamil and Hatcher, Allen and Lyczak, J T},
doi = {10.1016/S0079-8169(08)60506-1},
file = {:home/zack/Dropbox/Mendeley/Nadim, Hatcher, Lyczak/Pure and Applied Mathematics/Nadim, Hatcher, Lyczak - 1972 - Spectral Sequences.pdf:pdf;:home/zack/Dropbox/Mendeley/Nadim, Hatcher, Lyczak/Pure and Applied Mathematics/Nadim, Hatcher, Lyczak - 1972 - Spectral Sequences.pdf:pdf;:home/zack/Dropbox/Mendeley/Nadim, Hatcher, Lyczak/Pure and Applied Mathematics/Nadim, Hatcher, Lyczak - 1972 - Spectral Sequences.pdf:pdf},
isbn = {9780387683249},
issn = {00798169},
journal = {Pure Appl. Math.},
number = {January},
pages = {19--52},
title = {{Spectral Sequences}},
url = {https://www.math.cornell.edu/{~}hatcher/SSAT/SSATpage.html},
volume = {47},
year = {1972}
}
@article{Agoston1976a,
author = {Agoston, M K},
file = {:home/zack/Dropbox/Mendeley/Agoston/Unknown/Agoston - 1976 - Algebraic Topology Lectures.pdf:pdf},
title = {{Algebraic Topology Lectures}},
year = {1976}
}
@article{Muger2010,
abstract = {These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on k-linear categories with finite dimensional hom-spaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography.},
author = {M{\"{u}}ger, Michael},
journal = {Rev. la Union Mat. Argentina},
title = {{Tensor categories: A selective guided tour}},
url = {http://www.mendeley.com/research/tensor-categories-selective-guided-tour},
year = {2010}
}
@article{Consider2012,
file = {:home/zack/Dropbox/Mendeley/Unknown/Unknown/Unknown - 2012 - Cup Products.pdf:pdf},
pages = {1--12},
title = {{Cup Products}},
year = {2012}
}
@article{Ravenel2003,
abstract = {Since the publication of its first edition, this book has served as one of the few available on the classical Adams spectral sequence, and is the best account on the Adams-Novikov spectral sequence. This new edition has been updated in many places, especially the final chapter, which has been completely rewritten with an eye toward future research in the field. It remains the definitive reference on the stable homotopy groups of spheres. The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres. Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups. These topics are described in detail in Chapters 4 to 6. The revamped Chapter 7 is the computational payoff of the book, yielding a lot of information about the stable homotopy group of spheres. Appendices follow, giving self-contained accounts of the theory of formal group laws and the homological algebra associated with Hopf algebras and Hopf algebroids. The book is intended for anyone wishing to study computational stable homotopy theory. It is accessible to graduate students with a knowledge of algebraic topology and recommended to anyone wishing to venture into the frontiers of the subject.},
author = {Ravenel, Douglas C},
doi = {10.1090/chel/347},
file = {:home/zack/Dropbox/Mendeley/Ravenel/Unknown/Ravenel - 2003 - Complex Cobordism and Stable Homotopy Groups of Spheres Douglas.pdf:pdf},
isbn = {9780821829677},
pages = {395},
title = {{Complex Cobordism and Stable Homotopy Groups of Spheres Douglas}},
url = {http://books.google.co.kr/books?id=jeHtKa57OuIC{\&}printsec=frontcover{\&}dq=intitle:Complex+Cobordism+and+Stable+Homotopy+Groups+of+SpheresDouglas+C+Ravenel{\&}hl={\&}cd=1{\&}source=gbs{\_}api{\%}5Cnpapers3://publication/uuid/B468FC1A-8D7C-4B63-A586-217F561EFB8A},
year = {2003}
}
@article{Lukacs2003,
archivePrefix = {arXiv},
arxivId = {arXiv:1011.1669v3},
author = {Hatcher, Allen},
doi = {10.1002/ejoc.201200111},
eprint = {arXiv:1011.1669v3},
file = {:home/zack/Dropbox/Mendeley/Hatcher/Schriften des Forschungszentrum J�lich Reihe Energietechnik/Hatcher - 2003 - Vector Bundles and K Theory.pdf:pdf},
isbn = {9780132478663},
issn = {0196-6553},
journal = {Schriften des Forschungszentrum J�lich R. Energietechnik},
keywords = {Engineering,Materials,material},
number = {November},
pages = {2004},
pmid = {26840611},
title = {{Vector Bundles and K Theory}},
url = {https://www.faa.gov/data{\_}research/aviation/aerospace{\_}forecasts/media/FY2017-37{\_}FAA{\_}Aerospace{\_}Forecast.pdf},
volume = {21},
year = {2003}
}
@article{Hutchings2011a,
abstract = {Abstract The words “ spectral sequence ” strike fear into the hearts of many hardened mathematicians. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. This is an unfinished handout for my algebraic topology ... $\backslash$n},
author = {Hutchings, M},
file = {:home/zack/Dropbox/Mendeley/Hutchings/Unknown/Hutchings - 2011 - Introduction to spectral sequences.pdf:pdf},
pages = {1--13},
title = {{Introduction to spectral sequences}},
url = {http://www.math.uiuc.edu/{~}franklan/Hutchings{\_}ss.pdf{\%}5Cnpapers3://publication/uuid/24DC3CAA-57ED-409F-A65C-EA261AC25AFC},
volume = {1},
year = {2011}
}
@article{Ellis2010,
abstract = {We recall a description of the first non-vanishing homotopy group of a certain (n + 1)-ad of spaces and show how it yields group-theoretic formulae for homotopy and homology groups of several specific spaces. {\textcopyright} 2009 Elsevier Inc. All rights reserved.},
archivePrefix = {arXiv},
arxivId = {arXiv:0804.3581v1},
author = {Ellis, Graham and Mikhailov, Roman},
doi = {10.1016/j.aim.2009.11.003},
eprint = {arXiv:0804.3581v1},
file = {:home/zack/Dropbox/Mendeley/Ellis, Mikhailov/Advances in Mathematics/Ellis, Mikhailov - 2010 - A colimit of classifying spaces.pdf:pdf},
isbn = {0001-8708},
issn = {00018708},
journal = {Adv. Math. (N. Y).},
keywords = {Combinatorial group theory,Homotopy colimit,Homotopy group},
number = {6},
pages = {2097--2113},
title = {{A colimit of classifying spaces}},
volume = {223},
year = {2010}
}
@article{Hatcher,
abstract = {Short but advanced},
author = {Hatcher, Allen},
file = {:home/zack/Dropbox/Mendeley/Hatcher/Unknown/Hatcher - Unknown - Cohomology and Spectral Sequences.pdf:pdf},
pages = {381--402},
title = {{Cohomology and Spectral Sequences}},
url = {https://faculty.math.illinois.edu/{~}schenck/tapp.pdf}
}
@article{Vakil2008,
abstract = {Spectral sequences are a powerful book-keeping tool for proving things involving com-plicated commutative diagrams. They were introduced by Leray in the 1940's at the same time as he introduced sheaves. They have a reputation for being abstruse and difficult. It has been suggested that the name 'spectral' was given because, like spectres, spectral sequences are terrifying, evil, and dangerous. I have heard no one disagree with this interpretation, which is perhaps not surprising since I just made it up. Nonetheless, the goal of this note is to tell you enough that you can use spectral se-quences without hesitation or fear, and why you shouldn't be frightened when they come up in a seminar. What is different in this presentation is that we will use spectral sequence to prove things that you may have already seen, and that you can prove easily in other ways. This will allow you to get some hands-on experience for how to use them. We will also see them only in a " special case " of double complexes (which is the version by far the most often used in algebraic geometry), and not in the general form usually presented (filtered complexes, exact couples, etc.). See chapter 5 of Weibel's marvelous book for more detailed information if you wish. If you want to become comfortable with spectral sequences, you must try the exercises. For concreteness, we work in the category vector spaces over a given field. However, everything we say will apply in any abelian category, such as the category Mod A of A-modules.},
author = {Vakil, Ravi},
file = {:home/zack/Dropbox/Mendeley/Vakil/Unknown/Vakil - 2008 - Spectral Sequences Friend or Foe.pdf:pdf},
pages = {1--12},
title = {{Spectral Sequences : Friend or Foe ?}},
year = {2008}
}
@article{Peterson,
author = {Peterson, Eric},
file = {:home/zack/Dropbox/Mendeley/Peterson/Unknown/Peterson - Unknown - Computations with spectral sequences.pdf:pdf},
title = {{Computations with spectral sequences}}
}
@article{Hatcher2002,
author = {Hatcher, Allen},
doi = {10.2307/2374579},
file = {:home/zack/Dropbox/Mendeley/Hatcher/Algebraic Topology/Hatcher - 2002 - The Serre Spectral Sequence.pdf:pdf},
journal = {Algebr. Topol.},
pages = {1--67},
title = {{The Serre Spectral Sequence}},
year = {2002}
}
@article{Abanova,
author = {Abanov, A},
file = {:home/zack/Dropbox/Mendeley/Abanov/Unknown/Abanov - Unknown - Homotopy Groups Used in Physics.pdf:pdf},
pages = {35--40},
title = {{Homotopy Groups Used in Physics}},
volume = {0}
}
@article{Chow2006,
abstract = {The subject of spectral sequences has a reputation for being difficult for the beginner. Even GW Whitehead (quoted in John McCleary [4]) once remarked,“The machinery of spectral sequences , stemming from the algebraic work of Lyndon and Koszul, seemed ... $\backslash$n},
author = {Chow, Timothy Y},
file = {:home/zack/Dropbox/Mendeley/Chow/Notices of the AMS/Chow - 2006 - You could have invented spectral sequences.pdf:pdf},
isbn = {0002-9920},
issn = {0002-9920},
journal = {Not. AMS},
pages = {15--19},
title = {{You could have invented spectral sequences}},
url = {http://timothychow.net/spectral.pdf http://www.ams.org/notices/200601/fea-chow.pdf{\%}5Cnpapers3://publication/uuid/BB3D58F6-EE4C-4F23-B40E-02533806C685},
year = {2006}
}
@book{Hirsch1976a,
author = {Hirsch, Morris W C N - Book Collection (entrance floor) L I B 514.7 H I R O N SHELF},
booktitle = {Grad. texts Math.},
file = {:home/zack/Dropbox/Mendeley/Hirsch/Graduate texts in mathematics/Hirsch - 1976 - Differential topology.pdf:pdf},
isbn = {0387901485},
keywords = {Differential topology.},
pages = {x, 221 p},
title = {{Differential topology}},
year = {1976}
}
@article{Frankland2013,
author = {Frankland, Martin},
file = {:home/zack/Dropbox/Mendeley/Frankland/Unknown/Frankland - 2013 - Hurewicz Theorem and Applications.pdf:pdf},
title = {{Hurewicz Theorem and Applications}},
url = {http://www.home.uni-osnabrueck.de/mfrankland/Math527/Math527{\_}0325.pdf},
year = {2013}
}
@article{Greenberg2004,
author = {Schwab, Miriam and Futterer, Michael},
file = {:home/zack/Dropbox/Mendeley/Schwab, Futterer/Unknown/Schwab, Futterer - 2004 - Spectral Seuqneces.pdf:pdf},
title = {{Spectral Seuqneces}},
year = {2004}
}
@article{Feb2001,
archivePrefix = {arXiv},
arxivId = {arXiv:1103.5223v2},
author = {Feb, G T},
eprint = {arXiv:1103.5223v2},
file = {:home/zack/Dropbox/Mendeley/Feb/Unknown/Feb - 2001 - A Short Exposition of the Madsen-Weiss Theorem.pdf:pdf},
number = {1},
pages = {1--43},
title = {{A Short Exposition of the Madsen-Weiss Theorem}},
year = {2001}
}
@article{Fung2013,
author = {Fung, J U N H O U},
file = {:home/zack/Dropbox/Mendeley/Fung/Unknown/Fung - 2013 - Serre duality and applications.pdf:pdf},
pages = {1--35},
title = {{Serre duality and applications}},
year = {2013}
}
@book{Griffiths2013,
author = {Griffiths, Phillip and Morgan, John},
doi = {10.1007/978-1-4614-8468-4},
file = {:home/zack/Dropbox/Mendeley/Griffiths, Morgan/Unknown/Griffiths, Morgan - 2013 - Rational Homotopy Theory and Differential Forms.pdf:pdf},
isbn = {978-1-4614-8467-7},
title = {{Rational Homotopy Theory and Differential Forms}},
url = {http://link.springer.com/10.1007/978-1-4614-8468-4},
volume = {16},
year = {2013}
}
@article{Weston,
author = {Weston, T O M},
file = {:home/zack/Dropbox/Mendeley/Weston/Unknown/Weston - Unknown - The inflation-restriction sequence an introduction to spectral sequences.pdf:pdf},
pages = {1--8},
title = {{The inflation-restriction sequence : an introduction to spectral sequences}}
}
@article{Adams1960,
archivePrefix = {arXiv},
arxivId = {arXiv:0908.3724v4},
author = {Adams, J . F .},
eprint = {arXiv:0908.3724v4},
file = {:home/zack/Dropbox/Mendeley/Adams/Annals of Mathematics , Second Series , Vol . 72 , No . 1/Adams - 1960 - Annals of Mathematics On the Non-Existence of Elements of Hopf Invariant One.pdf:pdf},
journal = {Ann. Math. , Second Ser. , Vol . 72 , No . 1},
number = {1},
pages = {20--104},
title = {{Annals of Mathematics On the Non-Existence of Elements of Hopf Invariant One}},
volume = {72},
year = {1960}
}
@article{GrigorYan2015,
abstract = {We construct a cohomology theory on a category of finite digraphs (directed graphs), which is based on the universal calculus on the algebra of functions on the vertices of the digraph. We develop necessary algebraic technique and apply it for investigation of functorial properties of this theory. We introduce categories of digraphs and (undirected) graphs, and using natural isomorphism between the introduced category of graphs and the full subcategory of symmetric digraphs we transfer our cohomology theory to the category of graphs. Then we prove homotopy invariance of the introduced cohomology theory for undirected graphs. Thus we answer the question of Babson, Barcelo, Longueville, and Laubenbacher about existence of homotopy invariant homology theory for graphs. We establish connections with cohomology of simplicial complexes that arise naturally for some special classes of digraphs. For example, the cohomologies of posets coincide with the cohomologies of a simplicial complex associated with the poset. However, in general the digraph cohomology theory can not be reduced to simplicial cohomology. We describe the behavior of digraph cohomology groups for several topological constructions on the digraph level and prove that any given finite sequence of nonnegative integers can be realized as the sequence of ranks of digraph cohomology groups. We present also sufficiently many examples that illustrate the theory.},
author = {Grigor'Yan, Alexander and Lin, Yong and Muranov, Yuri and Yau, Shing Tung},
doi = {10.4310/AJM.2015.v19.n5.a5},
journal = {Asian J. Math.},
title = {{Cohomology of digraphs and (undirected) graphs}},
url = {http://www.mendeley.com/research/cohomology-digraphs-undirected-graphs},
year = {2015}
}
@article{Zeeman1957,
author = {Zeeman, E. C.},
doi = {10.1017/S0305004100031984},
issn = {0305-0041},
journal = {Math. Proc. Cambridge Philos. Soc.},
month = {jan},
number = {01},
pages = {57},
publisher = {Cambridge University Press},
title = {{A proof of the comparison theorem for spectral sequences}},
url = {http://www.journals.cambridge.org/abstract{\_}S0305004100031984},
volume = {53},
year = {1957}
}
@article{Hill2016,
abstract = {We show that the Kervaire invariant one elements $\theta$j ∈ $\pi$ 2 j+1 −2 S 0 exist only for j ≤ 6. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.},
author = {Hill, M. A. and Hopkins, M. J. and Ravenel, D. C.},
doi = {10.4007/annals.2016.184.1.1},
journal = {Ann. Math.},
title = {{On the nonexistence of elements of Kervaire invariant one}},
url = {http://www.mendeley.com/research/nonexistence-elements-kervaire-invariant-one},
year = {2016}
}
@article{Starkeya,
author = {Starkey, Ingrid},
file = {:home/zack/Dropbox/Mendeley/Starkey/Unknown/Starkey - Unknown - Homology theories.pdf:pdf},
pages = {1--12},
title = {{Homology theories}}
}
@misc{Jin,
author = {Jin, Alvin},
file = {:home/zack/Dropbox/Mendeley/Jin/Unknown/Jin - Unknown - Algebraic Topology Notes Fiber Bundles to Rational Homotopy Theory.pdf:pdf},
title = {{Algebraic Topology Notes: Fiber Bundles to Rational Homotopy Theory}},
url = {https://people.ucsc.edu/{~}anjin/courses/AT.pdf}
}
@article{Wang2010,
author = {Wang, Guozhen and Xu, Zhouli},
file = {:home/zack/Dropbox/Mendeley/Wang, Xu/Unknown/Wang, Xu - 2010 - A Survey of Computations of Homotopy Groups of Spheres and Cobordisms.pdf:pdf},
title = {{A Survey of Computations of Homotopy Groups of Spheres and Cobordisms}},
url = {http://math.mit.edu/{~}guozhen/homotopy groups.pdf},
year = {2010}
}
@article{Yin,
abstract = {This paper discusses basic properties of Eilenberg-MacLane spaces K(G, n), their cohomology groups and some classic applications. We construct K(G, n) using both CW complexes and classifying spaces. Two proofs are given to the basic fact that cohomology of a CW complex X has 1-1 correspondence with homotopy classes of maps from X into Eilenberg-MacLane spaces: a barehanded proof (original) and a categorical proof using loop spaces. We use it to demonstrate the connection between cohomology operations and cohomology groups of K(G, n)'s. Finally we use the technique of spectral sequence to compute the cohomology of some classes of Eilenberg-MacLane spaces, and apply it to the calculation $\pi$ 5 (S 3).},
author = {Yin, Xi},
file = {:home/zack/Dropbox/Mendeley/Yin/Construction/Yin - Unknown - On Eilenberg-MacLanes Spaces.pdf:pdf},
journal = {Construction},
pages = {1--16},
title = {{On Eilenberg-MacLanes Spaces}}
}
@article{Smith1957,
abstract = {When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied.The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors."This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.},
author = {Smith, P.A.},
doi = {10.1016/0016-0032(57)90941-9},
file = {:home/zack/Dropbox/Mendeley/Smith/Journal of the Franklin Institute/Smith - 1957 - Homological algebra.pdf:pdf},
isbn = {3540653783},
issn = {00160032},
journal = {J. Franklin Inst.},
number = {3},
pages = {259},
title = {{Homological algebra}},
url = {http://linkinghub.elsevier.com/retrieve/pii/0016003257909419},
volume = {263},
year = {1957}
}
@article{Strickland2008,
author = {Strickland, N P},
file = {:home/zack/Dropbox/Mendeley/Strickland/ReCALL/Strickland - 2008 - Spectral Sequences.pdf:pdf},
journal = {ReCALL},
pages = {1--9},
title = {{Spectral Sequences}},
year = {2008}
}
@article{Hutchings2011,
author = {Hutchings, Michael},
file = {:home/zack/Dropbox/Mendeley/Hutchings/Unknown/Hutchings - 2011 - Introduction to higher homotopy groups and obstruction theory.pdf:pdf},
pages = {1--38},
title = {{Introduction to higher homotopy groups and obstruction theory}},
year = {2011}
}